1

Vagueness• The Oxford Companion to Philosophy (1995):

“Words like smart, tall, and fat are vague since in most contexts of use there is no bright line separating them from not smart, not tall, and not fat respectively …”

2

Vagueness• Imprecision vs. Uncertainty:

The bottle is about half-full.

vs.

It is likely to a degree of 0.5 that the bottle is full.

3

Fuzzy Sets• Zadeh, L.A. (1965). Fuzzy Sets

Journal of Information and Control

4

Fuzzy Set DefinitionA fuzzy set is defined by a membership function that maps elements of a given domain (a crisp set) into values in [0, 1].A: U [0, 1]

A A

0

1

25

40

Age30

0.5young

5

Fuzzy Set Representation• Discrete domain:

high-dice score: {1:0, 2:0, 3:0.2, 4:0.5, 5:0.9, 6:1}

• Continuous domain:

A(u) = 1 for u[0, 25]A(u) = (40 - u)/15 for u[25, 40]A(u) = 0 for u[40, 150]

0

1

25

40

Age30

0.5

6

Fuzzy Set Representation• -cuts:

A = {u | A(u) }A = {u | A(u) } strong -cut

A0.5 = [0, 30]

0

1

25

40

Age30

0.5

7

Fuzzy Set Representation• -cuts:

A = {u | A(u) }A = {u | A(u) } strong -cut

A(u) = sup { | u A}

A0.5 = [0, 30]

0

1

25

40

Age30

0.5

8

Fuzzy Set Representation• Support:

supp(A) = {u | A(u) > 0} = A0+

• Core: core(A) = {u | A(u) = 1} = A1

• Height: h(A) = supUA(u)

9

Fuzzy Set Representation• Normal fuzzy set: h(A) = 1

• Sub-normal fuzzy set: h(A) 1

10

Membership Degrees• Subjective definition

• Voting model:Each voter has a subset of U as his/her own crisp definition of the concept that A represents.

A(u) is the proportion of voters whose crisp definitions include u.

A defines a probability distribution on the power set of U across the voters.

11

Membership Degrees• Voting model:

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10

1

2

3 x x4 x x x x x5 x x x x x x x x x6 x x x x x x x x x x

12

Fuzzy Subset Relations

A B iff A(u) B(u) for every uU

A is more specific than B

“X is A” entails “X is B”

13

Fuzzy Set Operations• Standard definitions:

Complement: A(u) = 1 A(u)

Intersection: (AB)(u) = min[A(u), B(u)]

Union: (AB)(u) = max[A(u), B(u)]

14

Fuzzy Set Operations• Example:

not young = young

not old = old

middle-age = not youngnot old

old = young

15

Fuzzy Numbers• A fuzzy number A is a fuzzy set on R:

A must be a normal fuzzy setA must be a closed interval for every (0, 1]supp(A) = A0+ must be bounded

16

Basic Types of Fuzzy Numbers

1

0

1

0

1

0

1

0

17

Basic Types of Fuzzy Numbers

1

0

1

0

18

Operations of Fuzzy Numbers• Extension principle for fuzzy sets:

f: U1...Un V

induces

g: U1...Un V ~~~

19

Operations of Fuzzy Numbers• Extension principle for fuzzy sets:

f: U1...Un V

induces

g: U1...Un V

[g(A1,...,An)](v) = sup{(u1,...,un) | v =

f(u1,...,un)}min{A1(u1),...,An(un)}

~~~

20

Operations of Fuzzy Numbers• EP-based operations:

(A + B)(z) = sup{(x,y) | z = x+y}min{A(x),B(y)}

(A B)(z) = sup{(x,y) | z = x-y}min{A(x),B(y)}

(A * B)(z) = sup{(x,y) | z = x*y}min{A(x),B(y)}

(A / B)(z) = sup{(x,y) | z = x/y}min{A(x),B(y)}

21

Operations of Fuzzy Numbers• EP-based operations:

+

1about 2 more or less 6 = about 2.about

3about 3

02 3 6

22

Operations of Fuzzy Numbers• Arithmetic operations on intervals:

[a, b][d, e] = {fg | a f b, d g e}

23

Operations of Fuzzy Numbers• Arithmetic operations on intervals:

[a, b][d, e] = {fg | a f b, d g e}

[a, b] + [d, e] = [a + d, b + e]

[a, b] [d, e] = [a e, b d]

[a, b]*[d, e] = [min(ad, ae, bd, be), max(ad, ae, bd, be)]

[a, b]/[d, e] = [a, b]*[1/e, 1/d] 0[d, e]

24

Operations of Fuzzy Numbers• Interval-based operations:

(A B)= A B

25

Possibility Theory• Possibility vs. Probability

• Possibility and Necessity

26

Possibility Theory• Zadeh, L.A. (1978). Fuzzy Sets as a Basis for a

Theory of PossibilityJournal of Fuzzy Sets and Systems

27

Possibility Theory• Membership degree = possibility degree

28

Possibility Theory• Axioms:

0 Pos(A) 1Pos() = 1 Pos() = 0Pos(A B) = max[Pos(A), Pos(B)]Nec(A) = 1 – Pos(A)

29

Possibility Theory• Derived properties:

Nec() = 1 Nec() = 0Nec(A B) = min[Nec(A), Nec(B)]max[Pos(A), Pos(A)] = 1min[Nec(A), Nec(A)] = 0Pos(A) + Pos(A) 1Nec(A) + Nec(A) 1Nec(A) Pos(A)

30

Possibility Theory

Probability Possibilityp: U [0, 1]Pro(A) = uAp(u)

r: U [0, 1]Pos(A) = maxuAr(u)

Normalization: uUp(u) = 1 maxuUr(u) = 1

Additivity: Pro(AB) = Pro(A) + Pro(B) Pro(AB)

Pos(AB) = max[Pos(A), Pos(B)]

Pro(A) + Pro(A) = 1 Pos(A) + Pos(A) 1

Total ignorance: p(u) = 1/|U| for every uU r(u) = 1 for every uU

Probability-possibility consistency principle: Pro(A) Pos(A)

31

Fuzzy Relations• Crisp relation:

R(U1, ..., Un) U1 ... Un

R(u1, ..., un) = 1 iff (u1, ..., un) R or = 0 otherwise

32

Fuzzy Relations• Crisp relation:

R(U1, ..., Un) U1 ... Un

R(u1, ..., un) = 1 iff (u1, ..., un) R or = 0 otherwise

• Fuzzy relation is a fuzzy set on U1 ... Un

33

Fuzzy Relations• Fuzzy relation:

U1 = {New York, Paris}, U2 = {Beijing, New York, London}R = very far

R = {(NY, Beijing): 1, ...}

NY ParisBeijing 1 .9NY 0 .7London .6 .3

34

Multivalued Logic• Truth values are in [0, 1]

• Lukasiewicz:a = 1 aa b = min(a, b)a b = max(a, b)a b = min(1, 1 a + b)

35

Fuzzy Logic• if x is A then y is B R(u, v) A(u) B(v)

x is A’ A’(u) ------------------------ ----------------------------------------

y is B’ B’(v)

36

Fuzzy Logic• if x is A then y is B

x is A’ ------------------------

y is B’

B’ = B + (A/A’)

37

Fuzzy Controller• As special expert systems

• When difficult to construct mathematical models

• When acquired models are expensive to use

38

Fuzzy ControllerIF the temperature is very highAND the pressure is slightly lowTHEN the heat change should be sligthly

negative

39

Fuzzy Controller

Controlledprocess

Defuzzificationmodel

Fuzzificationmodel

Fuzzy inference engine

Fuzzy rule base

actions

conditions

FUZZY CONTROLLER

40

Fuzzification

1

0x0

41

Defuzzification• Center of Area:

x = (A(z).z)/A(z)

42

Defuzzification• Center of Maxima:

M = {z | A(z) = h(A)}

x = (min M + max M)/2

43

Defuzzification• Mean of Maxima:

M = {z | A(z) = h(A)}

x = z/|M|

44

Exercises• In Klir’s FSFL: 1.9, 1.10, 2.11, 4.5, 5.1 (a)-(b), 8.6,

12.1.