Fuzzy logic Notes AI CSE 8th Sem

Artificial Intelligence Notes on Fuzzy Logic as according to CSVTU Syllabus for CSE 8th Sem

1. INTRODUCTION TO FUZZY LOGIC 1

2. Syllabus Fuzzy sets and crisp sets Intersections of Fuzzy sets, Union of Fuzzy sets, T he complement of Fuzzy sets. 2

3. Motivation The term “ fuzzy logic ” refers to a logic of approximation. Boolean logic assumes that every fact is either entirely true or false. Fuzzy logic allows for varying degrees of truth. Computers can apply this logic to represent vague and imprecise ideas, such as “ hot ” , “ tall ” or “ balding ” .

4. What is meant by fuzzy? Fuzzy (technical meaning) is imprecise, uncertain or unreliable knowledge, uncertain / noisy / incomplete Information, ambiguity (vague) or partial truth 4

5. Difference between imprecision and uncertainty Consider the following two situations : 1. John has at least two children and I am sure about it. 2. John has three children but I am not sure about it. In case 1, the number of children is imprecise but certain. In case 2, the number of children is precise but uncertain. 5

6. Uncertainty There is uncertainty that arises from ignorance, from various classes of randomness, from the inability to perform adequate measurements, from lack of knowledge, or from vagueness. 6

7. Types of Uncertainty 1. Stochastic uncertainty : It is the uncertainty towards the occurrence of a certain event. 2. Lexical uncertainty : It is the uncertainty lies in human languages like hot days, stable occurrence, a successful financial year and so on. 7

8. Ambiguity (vague) Food is hot. Here hot may be ‘spicy’ or ‘warm’ 8

9. Partial truth ??????????????? 9

10. World of information 10

11. Introduction Fuzzy Logic was initiated in 1965, by Dr. Lotfi A. Zadeh , professor for computer science at the university of California in Berkley. Fuzzy logic is a mathematical tool for dealing with uncertainty. It provides a technique to deal with imprecision and information granularity. The fuzzy theory provides a mechanism for representing linguistic constructs such as “many,” “low,” “medium,” “often,” “few.” 11

12. Classical set Classical sets are also called crisp set or nonfuzzy set. The traditional binary set theory describes crisp events, events that either do or do not occur. The crisp sets are sets without ambiguity in their membership. Example 1, for the set of integers, either an integer is even or it is not (it is odd). Example 2, However, either you are in the USA or you are not. 12

13. Example Lists: A = {apples, oranges, cherries, mangoes} A = {a 1 ,a 2 ,a 3 } A = {2, 4, 6, 8, …} Formulas: A = {x | x is an even natural number} A = {x | x = 2n, n is a natural number} Membership or characteristic function 13

14. Can they see each other? 14

15. Binary logic Vs. Fuzzy logic 15

16. Fuzzy sets It has an ability to classify elements into a continuous set using the concept of degree of membership. Fuzzy set is defined as a set whose elements have degrees of membership. The characteristics function or membership function not only gives 0 or 1 but can also give values between 0 and 1. Value 0-> non-membership Value 1->complete membership Value between 0 and 1-> degree of membership 16

17. Example of tumblers 17

18. Example There are following five tumblers, divided into two classes: full and empty. It is obvious: tumbler 1 belongs to the class full and tumbler 5 belongs to the class empty. Then tumblers 2, 3 and 4 belongs to which class? These tumblers are neither 100% full nor 100% empty. In other word we can say that tumbler 2 is 75% full, or 25% empty. 18

19. Example Now we define two sets: F and E. F is the set of all tumblers that belong to the class full. E is the set of all tumblers that belong to the class empty. 19

20. Graphical representation of sets 20

21. Fuzzy sets The sets F and E have some elements, which have not the full, but a partial membership. Such kind of non-crisp sets are called fuzzy sets. The set “all tumblers” that is here the basis of the fuzzy sets F and E, is called the Base set or universe of Discourse. 21

22. Fuzzy Logic Basically, Fuzzy Logic is a multivalued logic, that allows intermediate values to be defined between conventional evaluations like true/false, yes/no, high/low, etc. Fuzzy Logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth, i.e. truth values between “completely true” and “completely false”. Fuzzy Logic provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. 22

23. Fuzzy Logic Systems A system becomes a fuzzy system when its operations are entirely or partially governed by fuzzy logic or are based on fuzzy sets. A fuzzy logic system which accepts imprecise data and vague statements such as low, medium, high and provides decisions. 23

24. Fuzzy Logic Systems 24

25. Some Fuzzy Logic applications MASSIVE Created to help create the large-scale battle scenes in the Lord of the Rings films, MASSIVE is program for generating generating crowd-related visual effects

26. Applications of Fuzzy Logic Vehicle Control A number of subway systems, particularly in Japan and Europe, are using fuzzy systems to control braking and speed. One example is the Tokyo Monorail

27. Applications of Fuzzy Logic Appliance control systems Fuzzy logic is starting to be used to help control appliances ranging from rice cookers to small-scale microchips (such as the Freescale 68HC12)

28. Fuzzy Sets A fuzzy set A in X is expressed as a set of ordered pairs : Where, x is an element in X. 28

29. Membership function The membership function is a graphical representation of the magnitude of participation of each input. It associates a weighting with each of the inputs that are processed, define functional overlap between inputs, and ultimately determines an output response. The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. Once the functions are inferred, scaled, and combined, they are defuzzified into a crisp output which drives the system. 29

30. Continued…….. There are different membership functions associated with each input and output response. Some features to note are: SHAPE - triangular is common, but bell, trapezoidal, haversine and, exponential have been used. More complex functions are possible but require greater computing overhead to implement. HEIGHT or magnitude (usually normalized to 1) WIDTH (of the base of function), SHOULDERING (locks height at maximum if an outer function. Shouldered functions evaluate as 1.0 past their center) CENTER points (center of the member function shape) OVERLAP (N&Z, Z&P, typically about 50% of width but can be less).   30

31. Example For the tumbler example, fuzzy set can be represented as Full={(1,1), (2,0.75), (3,0.5), (4,0.25), (5,0)} 31

32. Fuzzy Sets with Discrete Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and nonordered ) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = “sensible number of children” X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)} 32

33. Fuzzy Sets with Cont. Universes Fuzzy set B = “about 50 years old” X = Set of positive real numbers (continuous) B = {(x, m B (x)) | x in X} 33

34. Alternative Notation A fuzzy set A can be alternatively denoted as follows: X is discrete X is continuous 34

35. Support(A) = {x  X |  A (x) > 0} The support of fuzzy set, is the crisp set of all points in the universe of discourse U such that membership function of A is non zero. Crossover(A) = {x  X |  A (x) = 0.5} The crossover point of fuzzy set , is the element in universe of discourse U at which its membership function is 0.5. Normal (A) = {x  X |  A (x) = 1} The fuzzy set is called normal if there is at least one element in the U where the membership function is 1. Characteristics of Fuzzy Sets 35

36. Characteristics of Fuzzy Sets The cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A,  A ( x ): card A =  A ( x 1 ) +  A ( x 2 ) + …  A ( x n ) = Σ  A ( x i ), for i =1.. n Example: Consider X = {1, 2, 3} and sets A and B A = 0.3/1 + 0.5/2 + 1/3; B = 0.5/1 + 0.55/2 + 1/3 card A = 1.8 card B = 2.05 36 Cardinality

37. Characteristics of Fuzzy Sets A fuzzy set A is empty, IF AND ONLY IF:  A ( x ) = 0,  x  X Example: Consider X = {1, 2, 3} and fuzzy set A = 0/1 + 0/2 + 0/3, then A is empty. 37 Empty Fuzzy Set

38. Characteristics of Fuzzy Sets An  -cut or  -level set of a fuzzy set A  X is an ORDINARY SET A   X , such that: A  ={  A ( x )  ,  x  X }. Strong alpha cut is defined as, A  ={  A ( x ) > ,  x  X }. Example: Consider X = {1, 2, 3} and set A = 0.3/1 + 0.5/2 + 1/3 then: A 0.5 = {2, 3}, A 0.1 = {1, 2, 3}, A 1 = {3}. 38 Alpha-Cut and Strong-alpha cut

39. Fuzzy singleton Fuzzy set whose support is a single point in X with µ A (x) =1 is called fuzzy singleton. 39 Characteristics of Fuzzy Sets

40. 40 Fuzzy set A is considered equal to a fuzzy set B , IF AND ONLY IF:  A ( x ) =  B ( x ),  x  X Example: A = 0.3/1 + 0.5/2 + 1/3, B = 0.3/1 + 0.5/2 + 1/3, therefore A = B. Equality Characteristics of Fuzzy Sets

41. Height The height of a fuzzy set ˜A is the maximum value of the membership function, i.e. max {μ(x)}. 41 Characteristics of Fuzzy Sets

42. Example 1 Let U is defined by X={1,2,3,4,5,6,7,8,9,10} & fuzzy set A={(1,0), (2,0.1), (3,0.2), (4,0.5), (5,0.3), (6,0.1),(7,0.0), (8,0.0), (9,0.0), (10,1)} Find the support, crossover and normal. 42

43. Solution 1 Support={2,3,4,5,6,10} Crossover point x=4 Normal point x=10 43

44. Logical Operations on Fuzzy set Fuzzy Intersection Fuzzy Union Complement 44

45. Logical Operations on Fuzzy set Union: The union the two sets A and B ( A  B ) can be defined by the membership function  U (x)   ( x )= max (   ( x ),   ( x )), x  X 45

46. Fuzzy Set Operations Union Union of 2 sets is comprised of those elements that belong to one or both sets.  A  B (X) = max (  A (x),  B (x))  x  X Example: Tall = {0/5, 0.2/5.5, 0.5/6, 0.8/6.5, 1/7} Short = {1/5, 0.8/5.5, 0.5/6, 0.2/6.5, 0/7}  tall  short = Attains its highest vales at the limits and lowest at the middle. Tall or short can mean not medium

47. Logical Operations on Fuzzy set Intersection: the intersection of two sets A and B ( A  B ) can be defined by the membership function   (x)   ( x )= min (   ( x ),   ( x )), x  X 47

48. Fuzzy Set Operations Intersection In classical set theory, intersection of 2 sets contains elements common to both. In fuzzy sets, an element may be partially in both sets.  A  B (X) = min (  A (x),  B (x))  x  X Example: Tall = {0/5, 0.2/5.5, 0.5/6, 0.8/6.5, 1/7} Short = {1/5, 0.8/5.5, 0.5/6, 0.2/6.5, 0/7}  tall  short = Tall and short can mean medium Highest at the middle and lowest at both end.

49. Logical Operations on Fuzzy set Complement: the complement of a fuzzy set A can be defined by the membership function mØ A ( x ) 49 mØ A ( x ) = 1 - m A ( x )

50. Fuzzy Set Operations Complementation (Not) Find complement ~A by using the following operation:  ~A (x) = 1 -  A (x) Example: Short = {1/5, 0.8/5.5, 0.5/6, 0.2/6.5, 0/7} Not short = { /5, /5.5, /6, /6.5, /7}

51. Example 2 Consider two fuzzy subsets of the set X , X = {a, b, c, d, e } referred to as A and B A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e} and B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e} Find out the Support, Cardinality, union, intersection and complement. 51

52. Solution 2 Support : supp ( A ) = {a, b, c, d } supp ( B ) = {a, b, c, d, e } Cardinality : card ( A ) = 1+0.3+0.2+0.8+0 = 2.3 card ( B ) = 0.6+0.9+0.1+0.3+0.2 = 2.1 Complement : A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}  A = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e} Union : A  B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e} Intersection : A  B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e} 52

53. Example 3 Suppose we have the following (discrete) fuzzy sets: A = 0.4/1 + 0.6/2 + 0.7/3 + 0.8/4 B = 0.3/1 + 0.65/2 + 0.4/3 + . 1/4 Represent A and B fuzzy sets graphically Calculate the of union of the set A and set B Calculate the intersection of the set A and set B Calculate the complement of the union of A and B 53

54. Solution (a) A B

55. Solution 3 ( b) The union of the fuzzy sets A and B = 0.4/1+0.65/2+0.7/3+0.8/4 (c) The intersection of the fuzzy sets A and B = 0.3/1+0.6/2+0.4/3+0.1/4 (c) The complement of the fuzzy set A = 0.6/1+0.4/2+0.3/3+0.2/4 55

56. Example 4 Given two fuzzy sets A and B a. Represent A and B fuzzy sets graphically b. Calculate the of union of the set A and set B c. Calculate the intersection of the set A and set B d. Calculate the complement of the union of A and B

57. Solution (a)

58. Solution 4 b c d

59. 59 Math Operations on Fuzzy Set kA = { k  A ( x ),  x  X } Let k =0.5, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then k A = {0.25/a, 0.15/b, 0.1/c, 0.5/d} A m = {  A ( x ) m ,  x  X } Let m =2, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then A m = {0.25/a, 0.09/b, 0.04/c, 1/d} …

60. 60 Example 5 There are two fuzzy subsets of the set X = {a, b, c, d, e }: A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e} and B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e }. Find the kA, A m and α -cut. K=0.5 and m=2

61. Solution 5 kA : for k =0.5 kA = {0.5/a, 0.15/b, 0.1/c, 0.4/d, 0/e} A m : for m =2 A a = {1/a, 0.09/b, 0.04/c, 0.64/d, 0/e} α -cut : A 0.2 = {a, b, c, d} A 0.3 = {a, b, d} A 0.8 = {a, d} A 1 = {a} 61

62. 62 Example 6 A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Draw the Fuzzy Graph of A and B Then, calculate the following: - Support, Core, Cardinality, and Complement for A and B independently - Union and Intersection of A and B - the new set C , if C = A 2 - the new set D , if D = 0.5  B - the new set E , for an alpha cut at A 0.5

63. 63 Solution 6 A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Support Supp(A) = {a, b, c, d} Supp(B) = {b, c, d, e} Core Core(A) = {c} Core(B) = {} Cardinality Card(A) = 0.2 + 0.4 + 1 + 0.8 + 0 = 2.4 Card(B) = 0 + 0.9 + 0.3 + 0.2 + 0.1 = 1.5 Complement Comp(A) = {0.8/a, 0.6/b, 0/c, 0.2/d, 1/e} Comp(B) = {1/a, 0.1/b, 0.7/c, 0.8/d, 0.9/e}

64. 64 Solution 6 cont.... A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Union A  B = {0.2/a, 0.9/b, 1/c, 0.8/d, 0.1/e} Intersection AB = {0/a, 0.4/b, 0.3/c, 0.2/d, 0/e} C=A 2 C = {0.04/a, 0.16/b, 1/c, 0.64/d, 0/e} D = 0.5  B D = {0/a, 0.45/b, 0.15/c, 0.1/d, 0.05/e} E = A 0.5 E = {c, d}

65. Algebraic Operations on Fuzzy Set Algebraic Product The product of two fuzzy sets in the same universe of discourse is the new fuzzy set A.B with a membership function that equal product of the membership function of A and the membership function of B.  A.B ( x ) ={ A ( x ) .  B ( x )| x  A , x  B } 65

66. Algebraic Operations on Fuzzy Set Multiplying a fuzzy set by a crisp number When a fuzzy set A is multiplied by a crisp number a, then its membership function is given by 66  a .A ( x ) = a A ( x )

67. Algebraic Operations on Fuzzy Set Cartesian product The Cartesian product of two fuzzy sets A & B is a fuzzy set C denoted by A X B and defined as 67 C=A X B=  c ( x )/( a,b ) |a  A, b  B  c ( C )=min(  A(a) ,  B(b) )

68. Algebraic Sum The Algebraic sum of two fuzzy sets A & B is a fuzzy set C denoted by A + B and defined as C=A + B=  c ( x )=  A ( x ) +  B ( x )-  A ( x ).  B ( x ) 68 Algebraic Operations on Fuzzy Set

69. Algebraic Operations on Fuzzy Set Bounded Sum The bounded sum of two fuzzy sets A and B in the universes X and Y with the membership functions μ A (x) and μ B (x) respectively is defined by 69 where the “+” sign is an arithmetic operator.

70. Algebraic Operations on Fuzzy Set Bounded product The bounded product of two fuzzy sets A and B in the universes X and Y with membership functions μ A (x) and μ B (x) respectively is defined as 70

71. Algebraic Operations on Fuzzy Set Bounded difference The bounded of difference of two fuzzy sets A and B is a fuzzy set C denoted by 71

72. Example 7 Let us consider two fuzzy sets A={(1,0.6),(2,1.0),(3,0.5),(4,0.3),(5,0.8) B={(2,0.5),(3,0.7)} Find out the Algebraic product, Cartesian product, Algebraic sum, Bounded Sum, Bounded difference. 72

73. Solution 7 73

74. Solution 7 cont……. 74

75. Solution 7 cont……. 75

76. Properties of Fuzzy sets Associative Property Commutative Property Distributive Property Idem Potency Identity Transitive Involution Demorgan’s Law 76

77. Properties of Fuzzy sets 77 Associative Property A∪ (B ∪ C) = (A ∪ B) ∪C or max [ μ A (X), max{ μ B (Y), μ C (Z)}] = max[ {max { μ A (x), μ B (Y)}, μ C (Z)] and A ∩ (B ∩ C) = (A ∩ B) ∩C or min [ μ A (X), min{ μ B (Y), μ C (Z)}] = min[ {min { μ A (x), μ B (Y)}, μ C (Z)]

78. Properties of Fuzzy sets 78 Commutative Property

79. Properties of Fuzzy sets 79 Distributive Property

80. Properties of Fuzzy sets 80 Idem Potency

81. Properties of Fuzzy sets 81 Identity

82. Properties of Fuzzy sets 82 Transitive

83. Properties of Fuzzy sets 83 Involution A’’=A

84. Properties of Fuzzy sets 84 Demorgan’s Law

85. Prove Prove that the De Morgan’s Laws hold for fuzzy sets that is 85

86. Solution 86

87. Solve 87

88. Solve 88

89. Solve 1. For the given fuzzy set 89 Prove the associative and the distributive property for the above given sets.

90. Solution To prove associative property 90

91. Solution To prove distribute property 91

92. Solve 92

93. Solve 3. Consider two fuzzy sets A∼ and B∼ find Complement, Union, Intersection, Difference, and De Morgan’s law. 93

94. Solve 4. Given the classical sets, A = {9, 5, 6, 8, 10} B = {1, 2, 3, 7, 9} C = {1, 0} Prove the classical set properties associativity and distributivity . 94

95. Solve 5. Consider, X = {a, b, c, d, e, f, g, h}. and the set A is defined as {a, d, f}. So for this classical set prove the identity property. 95

96. Solution 96

97. Solve 97

98. Solve 98