49:, when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a "type-2 fuzzy set". A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes.
137:—is much simpler than the mathematics that is needed for general type-2 fuzzy sets. So, the literature about interval type-2 fuzzy sets is large, whereas the literature about general type-2 fuzzy sets is much smaller. Both kinds of fuzzy sets are being actively researched by an ever-growing number of researchers around the world and have resulted in successful employment in a variety of domains such as robot control.
88:
319:", maps a type-1 fuzzy set into a number. There are many ways for doing this, e.g., compute the union of the fired-rule output fuzzy sets (the result is another type-1 fuzzy set) and then compute the center of gravity of the membership function for that set; compute a weighted average of the centers of gravity of each of the fired rule consequent membership functions; etc.
113:
125:
researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990s as a result of Jerry Mendel and his student's works on type-2 fuzzy sets and systems. Since then, more and more researchers around the world are writing articles about type-2 fuzzy sets and systems.
216:
331:
rules whose antecedents or consequents or both are uncertain. Just as standard deviation is widely used in probability and statistics to provide a measure of unpredictable uncertainty about a mean value, the type-reduced set can provide a measure of uncertainty about the crisp output of an interval type-2 FLS.
100:
For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for
330:
It is clear from Fig. 3 that there can be two outputs to an interval type-2 FLS—crisp numerical values and the type-reduced set. The latter provides a measure of the uncertainties that have flowed through the interval type-2 FLS, due to the (possibly) uncertain input measurements that have activated
307:
In most engineering applications of an FLS, a number (and not a fuzzy set) is needed as its final output, e.g., the consequent of the rule given above is "Rotate the valve a bit to the right." No automatic valve will know what this means because "a bit to the right" is a linguistic expression, and a
207:
Type-2 fuzzy sets are finding very wide applicability in rule-based fuzzy logic systems (FLSs) because they let uncertainties be modeled by them whereas such uncertainties cannot be modeled by type-1 fuzzy sets. A block diagram of a type-2 FLS is depicted in Fig. 3. This kind of FLS is used in fuzzy
322:
Things are somewhat more complicated for an interval type-2 FLS, because to go from an interval type-2 fuzzy set to a number (usually) requires two steps (Fig. 3). The first step, called "type-reduction", is where an interval type-2 fuzzy set is reduced to an interval-valued type-1 fuzzy set. There
348:
Of course, he did not mean that computers would actually compute using words—single words or phrases—rather than numbers. He meant that computers would be activated by words, which would be converted into a mathematical representation using fuzzy sets and that these fuzzy sets would be mapped by a
124:
Work on type-2 fuzzy sets languished during the 1980s and early-to-mid 1990s, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for
64:
In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. When the latter is done, the resulting type-2 fuzzy set is called a
120:
The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions (Fig. 2), a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set
247:
Fuzzy sets are associated with the terms that appear in the antecedents (IF-part) or consequents (THEN-part) of rules, and with the inputs to and the outputs of the FLS. Membership functions are used to describe these fuzzy sets, and in a type-1 FLS they are all type-1 fuzzy sets, whereas in an
91:
Figure 1. The membership function of a general type-2 fuzzy set is three-dimensional. A cross-section of one slice of the third dimension is shown. This cross-section, as well as all others, sits on the FOU. Only the boundary of the cross-section is used to describe the membership function of a
121:
mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets.
326:
The second step of Output
Processing, which occurs after type-reduction, is still called "defuzzification". Because a type-reduced set of an interval type-2 fuzzy set is always a finite interval of numbers, the defuzzified value is just the average of the two end-points of this interval.
223:
The following discussions, about the four components in Fig. 3 rule-based FLS, are given for an interval type-2 FLS, because to-date they are the most popular kind of type-2 FLS; however, most of the discussions are also applicable for a general type-2 FLS.
96:
The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig. 1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its "footprint of uncertainty"(FOU).
343:
CWW is a methodology in which the objects of computation are words and propositions drawn from a natural language. inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any
40:
so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word
349:
CWW engine into some other fuzzy set after which the latter would be converted back into a word. A natural question to ask is: Which kind of fuzzy set—type-1 or type-2—should be used as a model for a word? Mendel has argued, on the basis of
323:
are as many type-reduction methods as there are type-1 defuzzification methods. An algorithm developed by Karnik and Mendel now known as the "KM algorithm" is used for type-reduction. Although this algorithm is iterative, it is very fast.
45:, since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets,
291:
block. This is accomplished by first quantifying each rule using fuzzy set theory, and by then using the mathematics of fuzzy sets to establish the output of each rule, with the help of an inference mechanism. If there are
272:
block because it is fuzzy sets and not numbers that activate the rules which are described in terms of fuzzy sets and not numbers. Three kinds of fuzzifiers are possible in an interval type-2 FLS. When measurements are:
357:", that using a type-1 fuzzy set as a model for a word is scientifically incorrect. An interval type-2 fuzzy set should be used as a (first-order uncertainty) model for a word. Much research is underway about CWW.
714:
O. Salazar and J. Soriano, "Generating embedded type-1 fuzzy sets by means of convex combination," in
Proceedings of the 2013 IFSA World Congress NAFIPS Annual Meeting, Edmonton, Canada, Jun. 2013, pp. 51–56.
970:
Bibi, Youssouf, Omar
Bouhali, and Tarek Bouktir. "Petri type 2 fuzzy neural networks approximator for adaptive control of uncertain non-linear systems." IET Control Theory & Applications 11.17 (2017):
623:
O. Salazar, J. Soriano, and H. Serrano, "A short note on the centroid of an interval type-2 fuzzy set," in
Proceedings of IEEE 2012 Workshop on Engineering Applications (WEA), Bogota, Colombia, May 2012,
339:
Another application for fuzzy sets has also been inspired by Zadeh — "Computing with Words". Different acronyms have been used for "computing with words," e.g., CW and CWW. According to Zadeh:
308:
valve must be turned by numerical values, i.e. by a certain number of degrees. Consequently, the fired-rule output fuzzy sets have to be converted into a number, and this is done in the Fig. 3
723:
O. Salazar, and J. Soriano, "Convex combination and its application to fuzzy sets and interval-valued fuzzy sets I," Applied
Mathematical Sciences, vol. 9, no. 22, pp. 1061–1068, 2015
258:
Uncertain consequents—because when rules are obtained from a group of experts, consequents will often be different for the same rule, i.e. the experts will not necessarily be in agreement.
927:
Mo, Hong, Xuanming Zhao, and Fei-Yue Wang. "Application of
Interval Type-2 Fuzzy Sets in Unmanned Vehicle Visual Guidance." International Journal of Fuzzy Systems 21.6 (2019): 1661-1668.
583:
Hassanzadeh, Hamid Reza, et al. "An interval-valued fuzzy controller for complex dynamical systems with application to a 3-PSP parallel robot." Fuzzy sets and systems 235 (2014): 83-100.
732:
O. Salazar, and J. Soriano, "Convex combination and its application to fuzzy sets and interval-valued fuzzy sets II," Applied
Mathematical Sciences, vol. 9, no. 22, pp. 1069–1076, 2015
1005:"Type-2 Fuzzy Logic Controllers: Towards a New Approach for Handling Uncertainties in Real World Environments" by Hani Hagras, sponsored by the IEEE Computational Intelligence Society
296:
rules then the fuzzy input sets to the
Inference block will activate only a subset of those rules, where the subset contains at least one rule and usually way fewer than
918:
Dirik, Mahmut, Oscar
Castillo, and Adnan Fatih Kocamaz. "Visual-servoing based global path planning using interval type-2 fuzzy logic control." Axioms 8.2 (2019): 58.
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Noisy, but the noise is non-stationary, they are modeled as an interval type-2 fuzzy set (this latter kind of fuzzification cannot be done in a type-1 FLS).
659:
H. Bustince, "Indicator of inclusion grade for interval-valued fuzzy sets: Application to approximate reasoning based on interval-valued fuzzy sets,"
939:"A perceptual computing-based method to prioritize failure modes in failure mode and effect analysis and its application to edible bird nest farming"
741:
S. -M. Zhou, J. M. Garibaldi, R. I. John and F. Chiclana, "On constructing parsimonious type-2 fuzzy logic systems via influential rule selection,"
397:
Freeware MATLAB implementations, which cover general and interval type-2 fuzzy sets and systems, as well as type-1 fuzzy systems, are available at:
227:
Rules, that are either provided by subject experts or are extracted from numerical data, are expressed as a collection of IF-THEN statements, e.g.,
56:, while in type-2 fuzzy systems the membership function is fluctuating. A fuzzy set determines how input values are converted into fuzzy variables.
909:
Zarandi, MH Fazel, et al. "Designing a general type-2 fuzzy expert system for diagnosis of depression." Applied Soft
Computing 80 (2019): 329-341.
261:
Membership function parameters—because when those parameters are optimized using uncertain (noisy) training data, the parameters become uncertain.
688:
D. Wu and J. M. Mendel, "A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets,"
84:= 1, 2, ... . Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy.
544:
997:
1023:
832:
L. A. Zadeh, "From computing with numbers to computing with words—from manipulation of measurements to manipulation of perceptions,"
375:
133:
Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily
1002:"Introduction to Type-2 Fuzzy Sets and Systems" by Jerry Mendel, sponsored by the IEEE Computational Intelligence Society
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Words that are used in antecedents and consequents of rules—because words can mean different things to different people.
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rules. The inference is done one rule at a time. So, at the output of the Inference block, there will be one or more
287:
In Fig. 3, after measurements are fuzzified, the resulting input fuzzy sets are mapped into fuzzy output sets by the
105:
fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a
447:
900:
Castillo, Oscar, et al. "Review of recent type-2 fuzzy image processing applications." Information 8.3 (2017): 97.
646:
H. Wu and J. M. Mendel, "Uncertainty Bounds and Their Use in the Design of Interval Type-2 Fuzzy Logic Systems,"
208:
logic control, fuzzy logic signal processing, rule-based classification, etc., and is sometimes referred to as a
153:(a very widely used operation by practitioners of such sets, and also an important uncertainty measure for them)
938:
416:
Java libraries including source code for type-1, interval- and general type-2 fuzzy systems are available at:
68:
Zadeh didn't stop with type-2 fuzzy sets, because in that 1976 paper he also generalized all of this to type-
848:
L. A. Zadeh, "Toward human level machine intelligence—is it achievable? The need for a new paradigm shift,"
793:
D. Wu and J. M. Mendel, "Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets,"
672:
D. Wu and J. M. Mendel, "A Vector Similarity Measure for Interval Type-2 Fuzzy Sets and Type-1 Fuzzy Sets,"
780:
F. Liu and J. M. Mendel, "Aggregation Using the Fuzzy Weighted Average, as Computed by the KM Algorithms,"
380:
36:
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M. B. Gorzalczany, "A Method of Inference in Approximate Reasoning Based on Interval-Valued Fuzzy Sets,"
497:
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Tai, Kevin, et al. "Review of recent type-2 fuzzy controller applications." Algorithms 9.2 (2016): 39.
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F. Liu and J. M. Mendel, "Encoding words into interval type-2 fuzzy sets using an interval approach,"
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144:
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J. T. Rickard, J. Aisbett, G. Gibbon and D. Morgenthaler, "Fuzzy subsethood for type-n fuzzy sets,"
519:
L. A. Zadeh, "The Concept of a Linguistic Variable and Its Application to Approximate Reasoning–1,"
134:
53:
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An interval type-2 FLS lets any one or all of the following kinds of uncertainties be quantified:
472:
435:
An open source Matlab/Simulink Toolbox for Interval Type-2 Fuzzy Logic Systems is available at:
540:
17:
953:
116:
Figure 2. FOU for an interval type-2 fuzzy set. Many other shapes are possible for the FOU.
65:"general type-2 fuzzy set" (to distinguish it from the special interval type-2 fuzzy set).
386:
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411:
72:
fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the
46:
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interval type-2 FLS at least one membership function is an interval type-2 fuzzy set.
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application of fuzzy sets, because the FLS is designed to minimize an error function.
1012:
767:
Q. Liang and J. M. Mendel, "Interval Type-2 Fuzzy Logic Systems: Theory and Design,"
452:
140:
Formally, the following have already been worked out for interval type-2 fuzzy sets:
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Noisy measurements—because very often it is such measurements that activate the FLS.
87:
534:
436:
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Software supporting discrete interval type-2 fuzzy logic systems is available at:
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Noisy, but the noise is stationary, they are modeled as a type-1 fuzzy set; and,
268:
In Fig. 3, measured (crisp) inputs are first transformed into fuzzy sets in the
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D. Wu and J. M. Mendel, "Uncertainty measures for interval type-2 fuzzy sets,"
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32:
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Python library for interval type 2 fuzzy sets and systems is available at:
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IEEE Trans. on Circuits and Systems–1, Fundamental Theory and Applications
572:
Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions
487:
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Synthesizing an FOU from data that are collected from a group of subject
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J. M. Mendel, "Computing with Words: Zadeh, Turing, Popper and Occam,"
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general type-2 fuzzy set. It is shown filled-in for artistic purposes.
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this reason that an interval type-2 fuzzy set is sometimes called a
536:
Introduction To Type-2 Fuzzy Logic Control: Theory and Applications
592:
N. N. Karnik and J. M. Mendel, "Operations on Type-2 Fuzzy Sets,"
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86:
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N. N. Karnik and J. M. Mendel, "Centroid of a type-2 fuzzy set,"
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Python library for type 1 and type 2 fuzzy sets is available at:
891:(translation of Logik der Forschung), Hutchinson, London, 1959.
417:
996:
multi-media modules that can be accessed from the IEEE at:
189:
Firing intervals for an interval type-2 fuzzy logic system
533:
Jerry Mendel; Hani Hagras; Woei-Wan Tan (16 June 2014).
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J. M. Mendel, "Fuzzy Sets for Words: a New Beginning,"
365:
Type-2 fuzzy sets were applied in the following areas:
412:
http://dit2fls.com/projects/dit2fls-library-package/
819:L. A. Zadeh, "Fuzzy logic = computing with words,"
865:, St. Louis, MO, May 26–28, 2003, pp. 37–42.
76:in the logical progression from type-1 to type-
661:International Journal of Approximate Reasoning
574:, Prentice-Hall, Upper-Saddle River, NJ, 2001.
437:http://web.itu.edu.tr/kumbasart/type2fuzzy.htm
810:, vol. 16, pp. 1503–1521, December 2008.
797:, vol. 15, pp. 1145–1161, December 2007.
52:Type1 fuzzy systems are working with a fixed
8:
407:http://dit2fls.com/projects/dit2fls-toolbox/
315:In a type-1 FLS, output processing, called "
705:, Paper # 60101, New York City, May 2008.
277:Perfect, they are modeled as a crisp set;
878:, vol. 2, pp. 10–17, November 2007.
876:IEEE Computational Intelligence Magazine
850:IEEE Computational Intelligence Magazine
784:, vol. 16, pp. 1–12, February 2008.
425:https://github.com/carmelgafa/type2fuzzy
937:Chai K.C.; Tay K. M.; Lim C.P. (2016).
745:, vol.17, no.3, pp. 654–667, 2009.
650:, vol. 10, pp. 622–639, Oct. 2002.
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156:Other uncertainty measures [fuzziness,
852:, vol. 3, pp. 11–22, August 2008.
637:, vol. 177, pp. 5378–5393, 2007.
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399:http://sipi.usc.edu/~mendel/software
372:Video processing and computer vision
147:: union, intersection and complement
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676:, vol. 178, pp. 381–402, 2008.
614:, vol. 132, pp. 195–220, 2001.
596:, vol. 122, pp. 327–348, 2001.
431:https://github.com/Haghrah/PyIT2FLS
203:Interval type-2 fuzzy logic systems
663:, vol. 23, pp. 137–209, 2000.
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889:The Logic of Scientific Discovery
836:, vol. 4, pp. 105–119, 1999.
823:, vol. 4, pp. 103–111, 1996.
771:, vol. 8, pp. 535–550, 2000.
523:, vol. 8, pp. 199–249, 1975.
376:Failure Mode And Effect Analysis
183:Fuzzy rule ranking and selection
758:, vol. 21, pp. 1–17, 1987
1:
28:Type-2 fuzzy sets and systems
18:Type-2 Fuzzy Sets and Systems
821:IEEE Trans. on Fuzzy Systems
808:IEEE Trans. on Fuzzy Systems
795:IEEE Trans. on Fuzzy Systems
782:IEEE Trans. on Fuzzy Systems
769:IEEE Trans. on Fuzzy Systems
743:IEEE Trans. on Fuzzy Systems
648:IEEE Trans. on Fuzzy Systems
302:fired-rule fuzzy output sets
418:http://juzzy.wagnerweb.net/
195:Linguistic weighted average
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958:10.1016/j.asoc.2016.08.043
863:Proc. IEEE FUZZ Conference
448:Computational intelligence
410:DIT2FLS Library Package -
129:Interval type-2 fuzzy sets
1024:Logic in computer science
239:, then rotate the valve
107:second-order uncertainty
103:first-order uncertainty
946:Applied Soft Computing
756:Fuzzy Sets and Systems
594:Fuzzy Sets and Systems
381:Function approximation
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210:function approximation
192:Fuzzy weighted average
186:Type-reduction methods
168:and uncertainty bounds
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498:Random-fuzzy variable
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692:, to appear in 2009.
690:Information Sciences
674:Information Sciences
635:Information Sciences
612:Information Sciences
521:Information Sciences
478:Perceptual Computing
458:Fuzzy control system
335:Computing with words
219:Figure 3. Type-2 FLS
145:Fuzzy set operations
30:generalize standard
177:Embedded fuzzy sets
135:Interval arithmetic
54:membership function
473:Granular computing
405:DIT2FLS Toolbox -
241:a bit to the right
231:IF temperature is
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80:fuzzy sets, where
546:978-1-118-90144-1
310:Output Processing
180:Fuzzy set ranking
109:fuzzy set model.
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1019:Fuzzy logic
952:: 734–747.
887:K. Popper,
703:NAFIPS 2008
463:Fuzzy logic
351:Karl Popper
158:cardinality
1013:Categories
971:3130-3136.
504:References
174:Subsethood
171:Similarity
539:. Wiley.
493:Vagueness
483:Rough set
468:Fuzzy set
289:Inference
270:Fuzzifier
74:next step
488:Soft set
442:See also
393:Software
233:moderate
166:skewness
162:variance
151:Centroid
60:Overview
312:block.
37:systems
543:
942:(PDF)
43:fuzzy
541:ISBN
237:high
164:and
35:and
954:doi
1015::
950:49
948:.
944:.
841:^
681:^
601:^
555:^
512:^
420:.
304:.
160:,
960:.
956::
549:.
401:.
298:M
294:M
243:.
82:n
78:n
70:n
20:)
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