Solving Zadeh's Challenge Problems with the

Solving Zadeh’s Challenge Problems with the Application of RDM-Arithmetic Marcin Pluci´ nski() Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, ˙ lnierska 49, 71-062 Szczecin, Poland Zo [email protected]

Abstract. The paper presents a simple method of CwW that is based on RDM-models of intervals and on the multidimensional RDM-interval arithmetic. The method is explained on the example of popular Zadeh’s challenge problem known as ‘Balls in a box’ problem. On the base of presented calculations, the general methodology of solving CwW problems with the application of simplified RDM-models of quantifiers is formulated. Keywords: Computing with Words · Interval arithmetic · RDM-arithmetic · Zadeh’s challenge problems

1

Introduction

Computing with Words (CwW) is very important for decision-making, where very often, apart from numerical data, linguistic information provided by a problem expert is at disposal. Using linguistic and numerical information together enables a more effective decision-making than using only numerical one. Additionally, the linguistic information provided by the problem expert can be much more important and informative for the problem solving than numerical data, because accuracy of numerical data can be sometimes very low, although this data can make impression of a great precision. The idea of CwW, created by prof. Lotfi Zadeh, has been presented in numerous publications after 1990 [17–22, 24, 26]. From the very beginning CwW has been connected with fuzzy and interval arithmetic [3–5, 9]. CwW is rather a very difficult mathematical area and at present its development phase can be evaluated as a beginning state. This opinion is supported by the fact that practical possibilities of the present CwW are rather limited. Prof. Lotfi Zadeh has formulated many popular challenge problems. These problems can be found in his publications, eg. [25], and some of them are listed below. – Tall Swedes Most Swedes are tall. How many are short? What is the average height of Swedes? c Springer International Publishing Switzerland 2015 L. Rutkowski et al. (Eds.): ICAISC 2015, Part I, LNAI 9119, pp. 239–248, 2015. DOI: 10.1007/978-3-319-19324-3_22

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– Temperature Usually the temperature in my city is not very low and not very high. What is the average temperature? – Flight delay Usually most United Airlines flights from San Francisco leave on time. What is the probability that my flight will be delayed? – Balls in a box A box contains about 20 balls of various sizes. Most are large. What is the number of small balls? What is the probability that a ball drawn at random is neither small nor large? The above problems may seem very easy for non-specialists. However, it is not true. Many persons cannot solve them and such ability may be very important for autonomous thinking. If simple CwW examples cannot be solved, then, how more complex tasks could be? Solutions of Zadeh’s CwW-problems are in the scientific literature rather rare. Examples can be [1, 13–15]. These papers show how complicated are problems of autonomous thinking. They require considerable theoretical knowledge and each solution takes many pages. Therefore, industrial engineers or common economists can have difficulties with its application. The paper presents a less complicated method of CwW that is based on RDMmodels of intervals and on the multidimensional RDM-interval arithmetic. The method will be explained on the base of the last listed above problem, known as ‘Balls in a box’ problem.

2

RDM-Model of an Interval

The description of intervals with an application of RDM-models can be an interesting alternative to the commonly used interval arithmetic described by Moore [6, 7]. The author of the RDM-model concept is prof. Andrzej Piegat [10–12]. This concept allows to overcome many faults of the interval arithmetic such as: – the excess width effect, – the dependency problem, – difficulties in solving of even simplest interval equations, and others. These faults are described in greater detail in [2, 16]. Among others, they result from such features of the interval arithmetic as: nonexistence of additive and multiplicative inverse of intervals and failure to meet the distributivity law. The RDM-arithmetic has almost the same mathematical properties as conventional arithmetic [8]. As a result, complicated problems can be solved thanks to possibility of a formulas transformation. The RDM-arithmetic provides complete, multidimensional problem solutions from which various simplified representations as: a cardinality distribution, a span of a solution or an expected value of a solution can be derived.

Solving Zadeh’s Challenge Problem with the Application of RDM-Arithmetic

241

αx(x – x)

x

x

x

x

α=1

α=0

Fig. 1. The meaning of the RDM-variable α in the RDM-model of an interval

The RDM-arithmetic introduces an internal RDM-variable α ∈ [0, 1], which has a meaning of a relative-distance-measure (RDM), Fig. 1. An interval X can be described as: X = [x, x] = x + αx (x − x),

αx ∈ [0, 1] .

(1)

For example, the interval A = [1, 4] with use of the RDM-model has the form: A = 1 + 3αA ,

αA ∈ [0, 1] .

Thanks to RDM-variables, inside of intervals can also take part in calculations. One of the greatest advantages of the RDM-arithmetic, is a possibility of taking into account dependencies between intervals. In real problems, they are very common. Such dependencies can be taken into account by means of RDMvariables. For example in the classic interval arithmetic, as a result of subtracting two identical intervals we have: Y = X − X = [x, x] − [x, x] = [x − x, x − x]. If A = [1, 4]:

(2)

A − A = [1, 4] − [1, 4] = [−3, 3].

In the RDM-arithmetic, we can introduce two RDM-variables αx1 , αx2 ∈ [0, 1]: Y = X − X = (x + αx1 (x − x)) − (x + αx2 (x − x)) = (αx1 − αx2 )(x − x). (3) The result is a function of two RDM-variables αx1 and αx2 : Y = f (αx1 , αx2 ). We can calculate from it many parameters, eg. an interval lower and upper bound. The function value is the smallest for αx1 = 0 and αx2 = 1 – in that way we can get the lower bound of the resulting interval which is equal x − x. The function value is the greatest for αx1 = 1 and αx2 = 0 – in that way we can get the upper bound of the resulting interval which is equal x − x. If, however, we know that in the formula we have the same interval X, we can assume that αx1 = αx2 = αx . Then: Y = X − X = (x + αx (x − x)) − (x + αx (x − x)) = 0,

(4)

what is consistent with common sense, but unattainable for the classic interval arithmetic.

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Solution of ‘Balls in a box’ Problem

First, let us recall the problem that we want to solve. “A box contains about 20 balls of various sizes. Most are large. What is the number of small balls? What is the probability that a ball drawn at random is neither small nor large? [25]” First of all, let’s define the quantifier most. A proposal of such definition can be found in prof. Zadeh’s papers [19, 23] and the membership function of the quantifier is presented in Fig. 2a. The quantifier most describes the ratio of large balls to all balls in the box. In the paper, there will be used a simplified, interval form of it, Fig. 2b. A simplified form of the quantifier most can be described using the variable m which has the sense of part of the whole (PofW). RDM-model of the variable m has the form: 2 1 (5) m = + αm , αm ∈ [0, 1] . 3 3 The quantifier NOT (most ) will also be used in the solution. It can be interpreted as the minority of the whole, whereas the quantifier most has a sense of the majority of the whole. It can be defined as: N OT (m) = 1 − m =

1 1 − αm , 3 3

αm ∈ [0, 1] .

(6)

It should be noticed that the RDM-variable αm enables modeling of the dependency between quantifiers most and NOT (most ), and for their RDM-models holds: m + N OT (m) = 1 . (7) The dependency is illustrated in Fig. 3. The last question of the task suggests that there are large, small and other balls in the box. Let us define them as a medium-sized balls. Since we don’t μ

μ most

most 1

1

0

3/6 a)

5/6

1

0

4/6

PofW

1

PofW

b)

Fig. 2. The membership function of the linguistic quantifier most (a) and its simplified form (b)


NOT(most)

αm = 1 0

1/6

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most

αm = 0

αm = 0

1/3

2/3

αm = 1 5/6

PofW

1

Fig. 3. Illustration of the connection between RDM-models of quantifiers most and NOT (most)

have any information about amounts of small and medium balls, let’s create an additional variable s in the form of RDM-model: s = αs ,

αs ∈ [0, 1] ,

(8)

which determines what part of not large balls represent small ones. Respectively, the RDM-model: N OT (s) = 1 − αs ,

αs ∈ [0, 1] ,

(9)

determines what part of not large balls represent medium ones. The probability of the medium ball drawing can be calculated as: 1 1 − αm ·(1 − αs ) , αm ∈ [0, 1], αs ∈ [0, 1]. (10) p = N OT (m)·N OT (s) = 3 3 The plot of the function p = f (αm , αs ) is presented in Fig. 4. The value of the function of the possibility distribution of the medium ball drawing probability will be proportional to the contour lines length. From formula (10) we can find: αs = 1 −

3p 1 − αm

and

dαs −3 = , dαm (1 − αm )2 1

0.01

0.01 0.01

0.9 0.05

p

0.8 0.05 0.7

0.35

αs

0.25

0.4

0.15

0.3

0.1

0.15

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0.1 0.5 0 0.8

0.6

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0.2

0

0.2

0.1

αs

0

1

0.01

0.05

0.2

0.05

a)

0.05

0.5

0.2

1

0.01

0.1

0.6 0.3

0

0.3 0.1

0.25 0.2

0.2 0.3

0.4

0.15 0.5

0.1 0.6

0.7

0.8

0.9

1

αm b)

Fig. 4. The plot of the function p = f (αm , αs ) in the form of a surface (a) and in the form of contour lines (b)

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π(p) 1

1.8

0.9

1.6

0.8

1.4

0.7

1.2

0.6

1

0.5

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

Lp

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

p

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

p

b)

a)

Fig. 5. The plot of contour lines length as a function of the probability p (a) and the possibility distribution of the medium ball drawing probability (b)

and then the length of contour lines can be calculated as:

1−3p

Lp =

1+ 0

dαs dαm

2 dαm .

(11)

The plot of contour lines length as a function of the probability p is presented in Fig. 5a. For p = 0 the length is equal 2. In Fig. 5b, we can see the possibility distribution of the medium ball drawing probability. The expected value of the possibility equals approximately 0.1058. As an answer to the second question, we can say that the probability of the medium ball drawing can be known approximately as p ∈ [0, 13 ] and it can be expressed linguisticaly as about 0.1. Now, let’s calculate the number of small balls. As before, we can assume that the total amount of balls ‘about 20’ is described by the variable b in the form of RDM-model: (12) b = 16 + 8 · αb , αb ∈ [0, 1] . The number of small balls can be calculated as: 1 1 − αm · αs . n = b · N OT (m) · s = (16 + 8αb ) · 3 3

(13)

The variable n is the function of 3 RDM-variables: n = f (αm , αs , αb ). Fig. 6 presents 3D contour surfaces for integer values of n. The value of the function of the possibility distribution of the occurrence of certain amounts of small balls is proportional to the area of such created 3D contour surfaces, Fig. 6. From the formula (13) we can find: αb =

3 8n

(1 − αm )αs

−2,


245

n=0

n=4

αb

1

n=8 n=7

n=6

n=3

n=2

n=1

n=5

0.5

0 1 0.8 0.6 1

αs

0.8

0.4 0.6 0.2

0.4

αm

0.2 0

0

Fig. 6. 3D contour surfaces of the function n = f (αm , αs , αb ) for integer values of n

and then the area of each 3D contour surface can be calculated as: 2 2 dαb dαb 1+ + dαm dαs , An = dαm dαs D

(14)

where: D – area defined by the projection of a 3D contour surface onto the plane αm × αs . Examples of such areas for integer values of n are presented in Fig. 7. The area D is limited on the plane αm × αs by following lines: – for n < 5 31 : αm = 0,

3n , 16(1 − αm )

αs = 1,

αs =

αm = 0,

αs = 1,

– for n ≥ 5 31 : αs =

αs =

n , 8(1 − αm )

n . 8(1 − αm )

The plot of the area of 3D contour surfaces as a function of n is presented in Fig. 8a. In Fig. 8b, we can see the possibility distribution of the of the occurrence of certain amounts of small balls. Finally, we can say that the numer of small balls belongs to the interval [0, 8]. The expected value of the possibility equals approximately 2.149, but because the number of balls must be an integer value we can say that the most possible number equals 2.

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1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0

n=1 0

0.5

1

n=2

0

0

0.5

1

0

0.2

n=3 0

0.5

0

1

1

1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0

n=5 0

0.5

1

n=6

0

0

0.5

1

0

0.5

0.5

0

0.5

1

0.2

n=7 0

n=4 0

0

1

n=8 1

αm

Fig. 7. Exemplary areas of integration (14) for integer values of n An

π(n) 1

2.5

0.9 2

0.8 0.7

1.5

0.6 0.5

1

0.4 0.3

0.5

0.2 0.1

0

0

1

2

3

4

5

6

7

8

0

n

0

1

2

3

4

5

6

7

8

n

b)

a)

Fig. 8. The plot of the area of 3D contour surfaces as a function of n (a) and the possibility distribution of the occurrence of certain amounts of small balls (b)

The task can be also solved under the assumption that the RDM variable αb can only take values: 1 2 3 4 5 6 7 αb ∈ 0, , , , , , , , 1 8 8 8 8 8 8 8 to assure that the number of balls will always be integer – see (12). The expected value for such created possibility distribution function is equal approximately 1.8, so assuming that the number of balls must be an integer value we can also say that the most possible number equals 2.

4

Conclusions

The paper has described a CwW approach based on simplified RDM-models of linguistic quantifiers. Presented calculations are not complicated and obtained results are consistent with common sense.


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On the basis of the presented example, the following, general algorithm of solving CwW problems with the application of simplified RDM-model of quantifiers can be formulated. 1. Determine variables and RDM-models of linguistic quantifiers occurring in the problem. 2. Determine general formulas enabling the problem solving. 3. Find the solution as a function of RDM-variables. 4. Determine the possibility distribution of the solution and calculate the expected value. Such methodology is simple and can also be applied to other CwW problems.

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