シナジーのメタファー１

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C) defuzzification

　The exact value of the output variables that is assigned to various fuzzy sets is obtained by the defuzzification. That is, defuzzification is the transformation of ambiguous facts to concrete numbers and values. One calculates the centroid through numerical integration procedure by the Max/Min method.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

B) fuzzy inference
　Inference is always performed following the regulations of conjunction of a variable. The regulations of conjunction are also indicated as the processing rules.

(46) Syntax for the processing rules
If (premise 1) AND/OR (premise 2)
THEN (conclusion)

　For example, a status point from the childhood of Hans Castorp is described.
“Die sonderbare, halb träumerische, halb beängstigende Empfindung eines zugleich Ziehenden und Stehenden, eines wechselnden Bleibens, das Wiederkehr und schwindelige Einerleiheit war, - eine Empfindung, die ihm von früheren Gelegenheiten her bekannt war, und von der wieder berührt zu werden er erwartet und gewüscht hatte: sie war es zum Teil, um derentwillen ihm die Vorzeigung des stehend wandernden Erbstücks angelegen gewesen war.”（Der Zauberberg: 37)

　When his expectation to relate to the feeling is high AND suddenly the wish emerges THEN the irony is strong. In control engineering, it has proved itself AND conjunction of the minimum operator. A compensating operator (such as a gamma operator) can only be used on high-performance computers.
The sub-area of the membership function of the output variables is cut off by the Min/Max thresholds of each calculated membership value.

(47) μmiddle (Irony) = 0.2
μstrong (Irony) = 0.8

　Hence a configuration appears according to (48) below. The gray flat is obtained as the solution set (the index is used to represent the irony of Hans Castorp).

　The sub-area of the membership function of the output variables is multiplied by the Max/Prod method easily with each calculated membership values.

(48) μmiddle (Irony) = 0.2
μstrong (Irony) = 0.8

　Hence a configuration appears according to a solution set.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

The main areas of the fuzzy logic are control engineering and the decision making process. Here we will focus on control engineering, as the decision making process is very difficult to explain. How someone can control a process based on inaccurate values, shows that the control engineering application of fuzzy logic is seen as quite meaningful.
In fact, the fuzzy control can dictate a process that could not yet be controlled automatically. Fuzzy control does not need a mathematical process model, but rather inputs and outputs such as processing control on the basis of easy linguistic formation. Therefore, processes can also be controlled with hard or partially inaccessible process parameters.
Generally fuzzy control consists of fuzzification, fuzzy inference and defuzzification. (Hanamura 2005: 149)

A) Fuzzification
Through fuzzification, a given sharp value is assigned to a fuzzy set. The membership grade of the value to the fuzzy set is decided by the membership function as well, where it can also belong to multiple fuzzy sets. In the practice of control engineering, the membership function proves itself bit by bit with lineal process. For example, the grade (V0) of the expectation (see the next quotation) of Hans Castorp 7 (the index is used to represent his expectation). Hence we get the following equation.

(45)　μmiddle (V0) = 0.2
μhigh (V0) = 0.8

The expectation (V0) is 0.2 as the “middle” of the the fuzzy set and 0.8 as the “high”. In other words, a medium expectation for V0 is 20%, and a high expectation is 80%.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

　The membership grade of 36.0±0.4℃ to fuzzy set as “sick” is located in the range from 0.3 to 0.6. It proves to be practical to use the maximal value.

(41) μsick (36.0°C±0.4°C) = 0.6

　If another procedure should resolve a given problem better and can blend in the existent structure of fuzzy mathematics, the structure is designed to accept this.

(43)
μlow (36.0oC ± 0.4°C) = 0.1
μmiddle (36.0oC ±0.4oC) = max {0.5; 0.8} = 0.8
μhigh (36.0°C ± 0.4°C) = 0.5

　In this case, a medical thermometer becomes a central theme in the Magic Mountain.

“Joachim Ziemßen sagte, ≫es ist wohl auch bloß Konvention, daß ich hier vier
Striche zuviel habe auf meinem Thermometer! Aber wegen dieser fünf Striche muß ich mich hier herumräkeln und kann nicht Dienst machen, das ist eine ekelhafte Tatsache!≪
≫Hast du 37.5?≪，sagte Hans Castorp:
≫Es geht schon wieder herunter.≪ Und Joachim machte die Eintragung in eine
Tabelle. ≫Gestern abend waren es fast 38, das machte deine Ankunft.≪” (Der Zauberberg:96)

Hans Castorp also catches a fever again in another chapter.

“Nach Tische stieg das schimmernde Säulchen auf 37.7, verharrte abends, als der Patient nach den Erregungen und Neuigkeiten des Tages sehr müde war, auf 37.5, und zeigte in der nächsten Morgenfrühe gar nur auf 37, um gegen Mittag die gestrige Höhe wieder zu erreichen.” (Der Zauberberg: 247)

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

Hanamura (2005) describes what it is when the element by itself is fuzzy. That is, an element may also be not exactly 10 but “approximately 10” or “10±10%”. This occurs more frequently as it appears at first glance. All measurements aren’t absolute, but rather they are afflicted with measuring tolerance. Strictly speaking, the value indicated by a measuring device must not be taken unconditionally, but must always be provided with some measuring tolerance.
This procedure is obviously in measurement technology. Vividly, a fuzzy number can be seen as a small fuzzy set, an interval, whereby locating the middle number and determining the width based on measuring tolerances.
For example, let’s consider that a thermometer shows a body of 36℃, then the tolerance amounts to ca.±1%. Through this technique, a triangular process of the membership function has proved itself as practically convenient.

One knows the measured value (36℃) and the interval border that appears during the tolerance data. Poor measuring devices with larger tolerances leads to larger intervals, good measuring devices without any tolerance provides a unique discrete value. The vertical line about the measuring value (such as figure 38 below) indicates that a value taking tolerance into consideration is important.
How does one now determine the membership grade of a fuzzy number to a fuzzy set? The most plausible way is to select the maximal membership function value at the intersection of both membership functions. For example, a body temperature of 36.0℃±0.4℃ creates the below curve shape (figure 39).

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

The gamma operator is more important, because it mirrors the human feeling for compensatory AND appropriately.

(27) μAλB(x) = [μA(x)・μB(x)]1-Y. [1 - (1- μA (x))・(1 - μB (X))]Γ mitγ∈[0;1]

Similar to λ, it is determined by parameter Gamma, where the operator is located between pure AND, and pure OR.

(28) Gamma = Zero
μAλB(x) |γ=0 =μA(x)・μB(x)=μA and B

(29) Gamma = One
μAλB(x) γ=1 = 1 - (1 -μA(x)・(1 - μB(x))= 1 – [1 - μA(x) + μB(x) +μA(x)・μB(x)]
=μA(x) + μB(x) -μA(x)・μB(x)=μA OR B

(30) Graphically

AND OR
-----------------------------------------------------------------------------
Lambda =1 Lambda = 0
Gamma = 0 Gamma = 1
zero < ----------------------------------------------- > full
Compensation

The negation happens very simply. The assumption is certainly the normalized Description.

(31) Negation
μA/(x) = 1 -μA(x)

The modifiers (e.g. very, more or less) are seen as operators that influence a truth value but basically don’t change. They strengthen the features of the considered element or weakens them. The linguistic “very” can be mathematically obtained (as very much) by squaring the membership function. “More or less” can be mathematically described by the square root of the membership function.

(34) Modification
heated
not heated = 1 – heated patient
more or less patient= √patient
very patient = patient2
not very patient = 1 – very patient= 1 - patient2

Instead of the combination of a fuzzy set with a modifier, independent fuzzy sets can be also defined. Moreover, it has the advantage that the border of the separate sets can be individually determined.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

　By applying the minimum operator, the membership of the individual person to the set of the tolerant and strong man is as follows.

(23) For Hans Castorp
μduldsam and stark (Hans Castorp) = min (0.9; 0.5)= 0.5

(24) For Joachim Ziehmßen
μduldsam and stark (Joachim Ziehmßen) = min (0.6; 0.4) = 0.4

　(23) and (24) indicates that Hans Castorp belongs to the set of the tolerant and strong man, more than Joachim Ziehmßen.
　Following the classic logic, the set of the tolerant and strong men could be an empty set, since nobody fulfills both the features “tolerant” and “strong” simultaneously and completely. It is complemented by the fuzzy logic skillfully because a compensatory operator such as Lambda or Gamma must be located between pure AND (both features must be fulfilled) and pure OR (a feature must be fulfilled).

(25) μAλB(x) = λ・[μA(x)・μB(x)] + (1-λ)・[μA(x) +μB(x) - μA(x)・μB(x)] mit λ∈ [0;1]

(26) For λ= 0 man keeps an ODER operator
μAλB(x) |λ=0= μA(x) + μB(x) - μA(x)・μB(x)= μA OR B
For λ=1 man keeps an AND operator
μAλB(x) |λ=1=μA(x)・μB(x)=μA AND B

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

　According to Hanamura (2005) the difference between set- and logic operation is very important. Two fuzzy sets are completely related in set operations. A set appears at the end of the operation again. For example, a set of the tolerant child is related with the set of a moderate child. A set of tolerant and moderate appears at the end of the operation again. The characters of the element being considered are related in logic operations. An element appears with definite features. The feature “moderate” of a child (e.g. Hans Castorp) is related with the feature “moderate” of the child (e.g. AND relation). The element appears with the feature “tolerant-moderate” at the end.

(19) AND relation
μA UND B (x) = min{μA(x); μB(x)} as we say, minimum operator
μA UND B (x) =μA (x)・μB(x) as we say, product operator
μA UND B (x)= max{0; [μA(x) + μB(x)-1]}

(20) OR relation
μA ODER B (x) = max{μA(x); μB(x)} as we say, maximum operator
μA ODER B (x) = μA(x)+B(x) -μA(x)・μB(x)
μA ODER B (x) = min{1;[μA(x) +μB(x)]}

　When one considers the human logic and the way of thought, interestingly pure AND/ OR conjunction is rarely used. Usually a conjunction that appears somewhere in between AND/ OR conjunctions is used. In the Magic Mountain, one sees Hans Castorp as a hero and Joachim Ziemßen as a character. Hans Castorp is orphaned, tolerant and middle-size. Joachim Ziemßen is broad, big and careful.
　Here a tolerant and strong man is sought, whereby both the features “tolerant” and “strong” should be equally important at the selection time simplistically. Therefore, Hans Castorp can fulfill the feature “tolerant”, but only sometimes he can be seen as “strong”. Consequently, the membership could be assumed as 0.9 “tolerant” and 0.5 “strong”.

(21) μduldsam (Hans Castorp) =0.9
μstark (HansCastorp) = 0.5

Analogously it could be adapted for Joachim Ziemßen.

(22) μduldsam (Joachim Ziehmßen) = 0.6
μstark (Joachim Ziehmßen) = 0.4

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

(8) Unification set
A ⋃ B = {(x; μA ⋃B (x))} ∀x∈ G

(9) Intersecting set
A ⋂ B = {( x;μA ⋂ B(x))} ∀x ∈ G

(10) Distributive law
a. A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)
b. A ⋃ (B ⋂ C) = (A ⋃ B) ⋂ (A ⋃ C)

(11) Complement
A = {( x); μA (x)}∀x ∈ G with μA (x):=1 - μA (x) ∀x ∈ G

(12) Theorem von De Morgan
a. A⋃B// = A/⋂B/
b. A⋂B// = A/⋃B/

(13) Contained
A in B contained ⇔ μA(x) ≤μB(x) ∀x ∈ G

(14) Product of two sets
A・B = {(x; μA.B(x))} ∀x ∈ G with μA.B(x) := μA(x)・B(x) ∀x ∈ G
The product image of the normalized fuzzy set is commutative and associative.

(15) Sum
A+B = {(x;μA+B(x))} ∀x ∈ G mit μA+B(x) := μA(x) + μB(x) - μA(x).μB(x) ∀x∈G
The Sum image of the normalized fuzzy set is commutative and associative.

(16) Implication
When (A) then (B)
Mathematics: (x ∈ A) ⇒ (y ∈ B)
or short A⇒B
where (x), (y) are individual elements
X basic set to x, therefore x ∈ X
Y basic set to y, therefore y ∈ Y
A subset from X, therefore A ⊂ X
B subset from Y, therefore B ⊂Y

Zum Beispiel starben die Eltern von Hans Castorp in der kurzen Frist zwischen seinem fünften und siebenten Lebensjahr, zuerst die Mutter....
Da sein Vater sehr innig an seiner Frau gehangen hatte, auch seinerseits nicht der stärkste war, so wußte er nicht darüber hinwegzukommen. Sein Geist war verstört und geschmälert seitdem; in seiner Benommenheit beging er geschäftliche Fehler, so daß die Firma Castorp &Sohn empfindliche Verluste erlitt; im übernächsten Frühjahr holte er sich bei einer Speicherinspektion am windigen Hafen die Lungenentzündung, und da sein erschüttertes Herz das hohe Fieber nicht aushielt, so starb er trotz aller Sorgfalt.... (Der Zauberberg: 32)

x: momentary work
y: momentary health status
X: generally work = {easy, hard, boring, interesting,...}
Y: generally health status = {healthy, good, tired,...}
A: hard work = {too much, complicate,...}
B: bad health status = {painful, disordered, sick,...}

Implication: When the work is hard, then the body is disordered.
μhard (momentary) =1
μhard (momentary) = 0.8
μhard, disordered (momentary) = min (1; 0.8) = 0.8
The body is disordered by working too hard on a constant basis.
The assignment of the elements (x0) to the membership value μA(x0) mag be fuzzy. That is, the membership function μA(x) is fuzzy by itself. The case is called “ultra-fuzzy”. For example, one can determine whether a fixed child (Hans Castorp) is tolerant. To put it another way, to which extent does it belong to the fuzzy set “tolerant” (with parents, with father or mother, without parents etc.)?

(17) Ultra-fuzzy
An intervall [μA,1(x0); μA, 2 (x0)] is assigned to the value (x0) and μA(x0) represents “tolerant”. Here it is identified as to what extent a person (Typ 1) belongs to fuzzy set “tolerant”.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura

　A fuzzy set is the extension of the classical set theory. Hanamura (2005) exhibits it through linguistic variables such as “young” and “big” or by modifiers such as “very” or “most”. For example, how tall is Joachim Ziemßen? In the magic mountain Joachim Ziemßen could belong to the set of the bigger people than Hans Castorp. But how does he fulfill the feature of the fuzzy set? Here is a quantitative measure called the membership value and the membership function.

(2) μA (x) = 0.7
　t means that (x) has a membership value of 0.7 to the set A.
(3) a. μbig (J. Ziemßen)= 0.7
b. μbig (H. Castorp) = 0.3
　There are three kinds of the notations about the fuzzy set; Graphic notation (4), sum notation (5), and paired notation (6). A is a fuzzy set and xi is the element with the membership value μi.

(4) Graphic notation is the clearest and most often used (curve form is selected arbitrarily).

(5) The notation as a sum is used extremely rarely.
A =　μ1/x1 + μ2/x2+ … = �買ﾊi/xi∀x ∈ G
　This is only a possible notation of set (A). The membership value (μi) isn’t divided by the elements (xi), and the pairs (μi/ xi) are also not added.

(6) Notation as ordered pairs set
A = {( x1,μ1), (x2,μ2)，…} ∀x ∈ G
　When (G) is a selection of objects (x), then (A) is a fuzzy set with A = {(x; μA(x))|x ∈G}.
　The elements (xi) whose membership value is μi = 0 is not used to see the things clearly. Set operations and set combinations are also possible in the fuzzy set theory, because they include the sharp and classic set theory.

(7) A, B = fuzzy, normalized set
μA(x), μB(x) = Membership value of the elements (x) to the fuzzy set (A) or (B)
x = considered element
G = Set of all elements (x), therefore the primary set (sharp set involves all [x] completely)
min {...} = Minimum operator selects the minimum from the following curly bracket.
max {...} = Maximum operator selects the maximum from the following curly bracket.
∀ = Allquantor, read as “for all”.
∀x ∈ G means“for all elements x from the set G”.

花村嘉英（2005）「計算文学入門−Thomas Mannのイロニーはファジィ推論といえるのか？」より英訳 translated by Yoshihisa Hanamura