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Fuzzy Logic

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Fuzzy Logic Artificial Intelligence Chapter 9 Outline Crisp Logic Fuzzy Logic Fuzzy Logic Applications Conclusion Crisp Logic Crisp logic is concerned with absolutes ... – PowerPoint PPT presentation

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Title: Fuzzy Logic


1
Fuzzy Logic
  • Artificial Intelligence
  • Chapter 9

2
Outline
  • Crisp Logic
  • Fuzzy Logic
  • Fuzzy Logic Applications
  • Conclusion

3
Crisp Logic
  • Crisp logic is concerned with absolutes-true or
    false, there is no in-between.
  • Example
  • Rule
  • If the temperature is higher than 80F, it is
    hot otherwise, it is not hot.
  • Cases
  • Temperature 100F
  • Temperature 80.1F
  • Temperature 79.9F
  • Temperature 50F

Hot
Hot
Not hot
Not hot
4
Membership function of crisp logic
True
1
HOT
False
0
80F
Temperature
If temperature gt 80F, it is hot (1 or true) If
temperature lt 80F, it is not hot (0 or false).
5
Drawbacks of crisp logic
  • The membership function of crisp logic fails to
    distinguish between members of the same set.

6
Conception of Fuzzy Logic
  • Many decision-making and problem-solving tasks
    are too complex to be defined precisely
  • however, people succeed by using imprecise
    knowledge
  • Fuzzy logic resembles human reasoning in its use
    of approximate information and uncertainty to
    generate decisions.

7
Natural Language
  • Consider
  • Joe is tall -- what is tall?
  • Joe is very tall -- what does this differ from
    tall?
  • Natural language (like most other activities in
    life and indeed the universe) is not easily
    translated into the absolute terms of 0 and 1.

8
Fuzzy Logic
  • An approach to uncertainty that combines real
    values 01 and logic operations
  • Fuzzy logic is based on the ideas of fuzzy set
    theory and fuzzy set membership often found in
    natural (e.g., spoken) language.

9
Example Young
  • Example
  • Ann is 28, 0.8 in set Young
  • Bob is 35, 0.1 in set Young
  • Charlie is 23, 1.0 in set Young
  • Unlike statistics and probabilities, the degree
    is not describing probabilities that the item is
    in the set, but instead describes to what extent
    the item is the set.

10
Membership function of fuzzy logic
Fuzzy values
DOM Degree of Membership
Young
Old
Middle
1
0.5
0
25
40
55
Age
Fuzzy values have associated degrees of
membership in the set.
11
Crisp set vs. Fuzzy set
A traditional crisp set
A fuzzy set
12
Crisp set vs. Fuzzy set
13
Benefits of fuzzy logic
  • You want the value to switch gradually as Young
    becomes Middle and Middle becomes Old. This is
    the idea of fuzzy logic.
  • the label fuzzy logic is used in two senses
  • (a) narrow sense fuzzy logic is a logical system
  • (b) wide sense fuzzy logic is coextensive with
    fuzzy set theory
  • Applications
  • Control
  • Business
  • Finance

14
Fuzzy in control
  • Control Rules
  • 1.If (speed is low) and (shift is high) then (-3)
  • 2.If (speed is high) and (shift is low) then (3)
  • 3.If (throtis low) and (speed is high) then (3)
  • 4.If (throtis low) and (speed is low) then (1)
  • 5.If (throtis high) and (speed is high) then (-1)
  • 6.If (throtis high) and (speed is low) then (-3)

15
Fuzzy Set Operations
  • Fuzzy union (?) the union of two fuzzy sets is
    the maximum (MAX) of each element from two sets.
  • E.g.
  • A 1.0, 0.20, 0.75
  • B 0.2, 0.45, 0.50
  • A ? B MAX(1.0, 0.2), MAX(0.20, 0.45),
    MAX(0.75, 0.50)
  • 1.0, 0.45, 0.75

16
Fuzzy Set Operations
  • Fuzzy intersection (?) the intersection of two
    fuzzy sets is just the MIN of each element from
    the two sets.
  • E.g.
  • A 1.0, 0.20, 0.75
  • B 0.2, 0.45, 0.50
  • A ? B MIN(1.0, 0.2), MIN(0.20, 0.45),
    MIN(0.75, 0.50) 0.2, 0.20, 0.50

17
Fuzzy Set Operations
  • The complement of a fuzzy variable with DOM x is
    (1-x).
  • Complement ( _c) The complement of a fuzzy set
    is composed of all elements complement.
  • Example.
  • A 1.0, 0.20, 0.75
  • Ac 1 1.0, 1 0.2, 1 0.75 0.0, 0.8,
    0.25

18
Crisp Relations
  • Ordered pairs showing connection between two
    sets
  • (a,b) a is related to b
  • (2,3) are related with the
    relation lt
  • Relations are set themselves
  • lt (1,2), (2, 3), (2, 4), .
  • Relations can be expressed as matrices

lt 1 2
1 ? ?
2 ? ?

19
Fuzzy Relations
  • Triples showing connection between two sets
  • (a,b,) a is related to b with
    degree
  • Fuzzy relations are set themselves
  • Fuzzy relations can be expressed as matrices


20
Fuzzy Relations Matrices
  • Example Color-Ripeness relation for tomatoes

R1(x, y) unripe semi ripe ripe
green 1 0.5 0
yellow 0.3 1 0.4
Red 0 0.2 1
21
Where is Fuzzy Logic used?
  • Fuzzy logic is used directly in very few
    applications.
  • Most applications of fuzzy logic use it as the
    underlying logic system for decision support
    systems.

22
Fuzzy Expert System
  • Fuzzy expert system is a collection of membership
    functions and rules that are used to reason about
    data.
  • Usually, the rules in a fuzzy expert system are
    have the following form
  • if x is low and y is high then z is medium

23
Operation of Fuzzy System
Crisp Input
Fuzzification
Input Membership Functions
Fuzzy Input
Rule Evaluation
Rules / Inferences
Fuzzy Output
Defuzzification
Output Membership Functions
Crisp Output
24
Fuzzification
  • Two Inputs (x, y) and one output (z)
  • Membership functions
  • low(t) 1 - ( t / 10 )
  • high(t) t / 10

1
0.68
Low
High
0.32
0
t
Crisp Inputs
X0.32
Y0.61
Low(x) 0.68, High(x) 0.32,
Low(y) 0.39, High(y) 0.61
25
Create rule base
  • Rule 1 If x is low AND y is low Then z is high
  • Rule 2 If x is low AND y is high Then z is low
  • Rule 3 If x is high AND y is low Then z is low
  • Rule 4 If x is high AND y is high Then z is high

26
Inference
  • Rule1 low(x)0.68, low(y)0.39 gt
    high(z)MIN(0.68,0.39)0.39
  • Rule2 low(x)0.68, high(y)0.61 gt
    low(z)MIN(0.68,0.61)0.61
  • Rule3 high(x)0.32, low(y)0.39 gt
    low(z)MIN(0.32,0.39)0.32
  • Rule4 high(x)0.32, high(y)0.61 gt
    high(z)MIN(0.32,0.61)0.32

Rule strength
27
Composition
  • Low(z) MAX(rule2, rule3) MAX(0.61, 0.32)
    0.61
  • High(z) MAX(rule1, rule4) MAX(0.39, 0.32)
    0.39

1
Low
High
0.61
0.39
0
t
28
Defuzzification
  • Center of Gravity

1
Low
High
Center of Gravity
0.61
0.39
0
t
Crisp output
29
Fuzzy Relations
30
Fuzzy Relations
  • Ordered pairs showing connection between two sets
  • Relations are sets themselves
  • Expressed as matrices

31
Fuzzy Relations
  • Value of the membership function, mR(x, y), for
    an element (x, y) of the relation R is the value
    at row x and column y in the relational matrix
  • Shows degree of correspondence between
    x-qualities (color) and y-qualities (ripeness)

32
Fuzzy Relations Matrices
  • Color ripeness relation for tomatoes

33
Fuzzy Relations Matrices
  • Ripeness - taste relation for tomatoes

34
Fuzzy Relations Matrices
  • Color - taste relation for tomatoes

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