Fuzzy LogicBy Manoj Harsule
Overview•   A Little History•   Fuzzy Logic – A Definition•   Fuzzy set theory•   Introduction to fuzzy set•   Fuzzy Relat...
A little History In the 1960’s Lotfi A. Zadeh Ph.D,. University of  California, Berkeley, published an obscure paper on  ...
WHAT IS FUZZY LOGIC?   Definition of fuzzy     Fuzzy   – “not clear, distinct, or not precise;      uncertain”   Defini...
TRADITIONAL REPRESENTATION OFLOGICSlow (Low)         Fast (High)Speed = 0        Speed = 1
FUZZY LOGICREPRESENTATION                    Slowest For every         [ 0.0 – 0.25 ]  problem           Slow  must      ...
Introduction toFuzzy Set TheoryFuzzy Sets
Types of Uncertainty• Stochastic uncertainty  – E.g., rolling a dice• Linguistic uncertainty  – E.g., low price, tall peop...
Crisp or Fuzzy Logic• Crisp Logic  – A proposition can be true or false only.    • Ajay is a student (true)    • Smoking i...
Crisp Sets• Classical sets are called crisp sets  – either an element belongs to a set or    not, i.e.,     x∈ A          ...
Crisp Sets  P : the set of all people.  Y : the set of all young people.             P                                    ...
Crisp sets   µ A ( x) ∈ { 0,1}Fuzzy Sets     µ A ( x) ∈ [0,1]               ExampleµYoung ( y )          1                ...
Definition:Fuzzy Sets and MembershipFunctions  U : universe of discourse.If U is a collection of objects denoted generical...
Example (Discrete    Universe) U = {1, 2,3, 4,5, 6, 7,8}  # courses a student may take in a semester.      (1, 0.1) (2, 0...
Example (Discrete   Universe)U = {1, 2,3, 4,5, 6, 7,8}                       # courses a student may                      ...
Example (Continuous Universe)  U : the set of positive real numbers                               possible ages      B = {...
Alternative Notation          A = { ( x, µ A ( x)) x ∈ U } U : discrete universe                      A=    ∑µ            ...
Membership Functions (MF’s)• A fuzzy set is completely characterized by  a membership function.  – a subjective measure.  ...
Fuzzy Partition• Fuzzy partitions formed by the linguistic  values “young”, “middle aged”, and “old”:
Introduction toFuzzy Set Theory   Set-Theoretic    Operations
Set-TheoreticOperations• Subset  A ⊆B ⇔µA ( x ) ≤ µ ( x ), ∀ ∈                    B        x U• Complement  A = U − A ⇔ µA...
Set-TheoreticOperations A⊂ B                A                  A∩ B           A∪ B
PropertiesInvolution         A=A                                                       De Morgan’s laws                   ...
Properties• The following properties are invalid  for fuzzy sets:  – The laws of contradiction            A∩ A = ∅  – The ...
Other Definitions for SetOperations• Union µ A∪ B ( x) = min ( 1, µ A ( x) + µ B ( x) )• Intersection  µ A∩ B ( x) = µ A (...
Other Definitions for SetOperations• Unionµ A∪ B ( x ) = µ A ( x ) + µ B ( x ) − µ A ( x ) µ B ( x )• Intersection µ A∩ B ...
GeneralizedUnion/Intersection• Generalized Intersection      t-norm• Generalized Union      t-conorm
T-Norm                           Or called triangular norm.         T :[0,1] × [0,1] → [0,1]1. Symmetry        T ( x, y ) ...
T-Conorm Or called s-norm.         S :[0,1] × [0,1] → [0,1]1. Symmetry        S ( x, y ) = S ( y , x )2. Associativity   S...
Fuzzy Relations Review Fuzzy Relations
R ⊆ A×BBinary Relation (R)                                               b1         a1                                    ...
The Real-Life Relation• x is close to y   – x and y are numbers• x depends on y   – x and y are events• x and y look alike...
Fuzzy RelationsA fuzzy relation R is a 2D MF: R = { ( ( x, y ), µ R ( x, y ) ) | ( x, y ) ∈ X × Y }
R = { ( ( x, y ), µ R ( x, y ) ) | ( x, y ) ∈ X × Y } Example (Approximate Equal)X = Y = U = {1, 2,3, 4,5}                ...
Max-Min CompositionX         Y           Z                              R: fuzzy relation defined on X and Y.             ...
µ S o R (x, y ) = max v min ( µ R ( x, v), µ S (v, y ) )    ExampleR   a   b    c   d                                 S α ...
Max-min composition is not mathematically                    tractable, therefore other compositions such as              ...
Dimension ReductionProjection                     RRY =  R ↓ Y                         RX =  R ↓ X                  ...
Dimension ReductionProjection                                RRY =  R ↓ Y                                   RX =  R ↓...
Dimension ExpansionCylindrical ExtensionA : a fuzzy set in X.C(A) = [A↑X×Y] : cylindrical extension of A. C ( A) = ∫      ...
Types of Fuzzy Relations                              R ( x, x) = 1 for all x ∈ X• Reflexive  – Irreflexive      R ( x, x)...
Types of Fuzzy Relations• Transitive (max-min transitive)  R ( x, z ) ≥ max min[ R ( x, y ), R ( y , z )] for all x,z ∈X  ...
Types of Fuzzy Relations• Transitive Closure  – Crisp: Transitive relation that contains    R(X,X) with fewest possible me...
Types of Fuzzy Relations• Fuzzy Equivalence or Similarity  Relation  – Reflexive, symmetric, and transitive  – Decompositi...
Types of Fuzzy Relations• Fuzzy Compatibility or Tolerance Relation  – Reflexive and symmetric  – Maximal compatibility cl...
Bibliography• J. R. Jang, C. Sun, E. Mizutani,  “Neuro-Fuzzy and Soft Computing: A  Computational Approach to Learning  an...
Introduction to Artificial Intelligence
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Introduction to Artificial Intelligence

  1. 1. Fuzzy LogicBy Manoj Harsule
  2. 2. Overview• A Little History• Fuzzy Logic – A Definition• Fuzzy set theory• Introduction to fuzzy set• Fuzzy Relations
  3. 3. A little History In the 1960’s Lotfi A. Zadeh Ph.D,. University of California, Berkeley, published an obscure paper on fuzzy sets . His unconventional theory allowed for approximate information and uncertainty when generating complex solutions; a process that previously did not exist. Fuzzy Logic has been around since the mid 60’s but was not readily excepted until the 80’s and 90’s. Although now prevalent throughout much of the world, China, Japan and Korea were the early adopters
  4. 4. WHAT IS FUZZY LOGIC? Definition of fuzzy  Fuzzy – “not clear, distinct, or not precise; uncertain” Definition of fuzzy logic A form of knowledge representation suitable for notions that cannot be defined precisely, but which depend upon their contexts.
  5. 5. TRADITIONAL REPRESENTATION OFLOGICSlow (Low) Fast (High)Speed = 0 Speed = 1
  6. 6. FUZZY LOGICREPRESENTATION Slowest For every [ 0.0 – 0.25 ] problem Slow must [ 0.25 – 0.50 ] represent in Fast terms of [ 0.50 – 0.75 ] fuzzy sets. Fastest [ 0.75 – 1.00 ]
  7. 7. Introduction toFuzzy Set TheoryFuzzy Sets
  8. 8. Types of Uncertainty• Stochastic uncertainty – E.g., rolling a dice• Linguistic uncertainty – E.g., low price, tall people, young age• Informational uncertainty – E.g., credit worthiness, honesty
  9. 9. Crisp or Fuzzy Logic• Crisp Logic – A proposition can be true or false only. • Ajay is a student (true) • Smoking is healthy (false) – The degree of truth is 0 or 1.• Fuzzy Logic – The degree of truth is between 0 and 1. • Raj is young (0.3 truth) • Amol is smart (0.9 truth)
  10. 10. Crisp Sets• Classical sets are called crisp sets – either an element belongs to a set or not, i.e., x∈ A or x∉ A• Member Function of crisp set 0 x∉ A µ A ( x) =  µ A ( x) ∈ { 0,1} 1 x ∈ A
  11. 11. Crisp Sets P : the set of all people. Y : the set of all young people. P Y YYoung = { y y = age( x) ≤ 25, x ∈ P} µYoung ( y ) 1 25 y
  12. 12. Crisp sets µ A ( x) ∈ { 0,1}Fuzzy Sets µ A ( x) ∈ [0,1] ExampleµYoung ( y ) 1 y
  13. 13. Definition:Fuzzy Sets and MembershipFunctions U : universe of discourse.If U is a collection of objects denoted generically by x, thena fuzzy set A in U is defined as a set of ordered pairs: A = { ( x, µ A ( x)) x ∈ U } membership function µ A : U → [0,1]
  14. 14. Example (Discrete Universe) U = {1, 2,3, 4,5, 6, 7,8} # courses a student may take in a semester.  (1, 0.1) (2, 0.3) (3, 0.8) (4,1)  A=  #appropriate  (5, 0.9) (6, 0.5) (7, 0.2) (8, 0.1)  courses taken 1µA ( x ) 0.5 0 2 4 6 8 x : # courses
  15. 15. Example (Discrete Universe)U = {1, 2,3, 4,5, 6, 7,8} # courses a student may take in a semester.  (1,0.1) (2,0.3) (3,0.8) (4,1)  appropriateA=    (5,0.9) (6,0.5) (7,0.2) (8,0.1)  # courses taken Alternative Representation: A = 0.1/ 1 + 0.3 / 2 + 0.8 / 3 + 1.0 / 4 + 0.9 / 5 + 0.5 / 6 + 0.2 / 7 + 0.1/ 8
  16. 16. Example (Continuous Universe) U : the set of positive real numbers possible ages B = { ( x, µ B ( x)) x ∈U } 1 about 50 years oldµ B ( x) = 4  x − 50  1.2 1+  ÷ 1  5  0.8 µ B ( x) 0.6Alternative 0.4Representation: 0.2 0 B=∫ 1 x 0 20 40 60 80 100 ( ) 4 x : age R + 1+ x −50 5
  17. 17. Alternative Notation A = { ( x, µ A ( x)) x ∈ U } U : discrete universe A= ∑µ xi ∈U A ( xi ) / xiU : continuous universe A = ∫ µ A ( x) / x UNote that ∑ and integral signs stand for the union of membership grades; “/ ” stands for a marker and does not imply division.
  18. 18. Membership Functions (MF’s)• A fuzzy set is completely characterized by a membership function. – a subjective measure. – not a probability measure. “tall” in Asia Membership 1 value “tall” in USA “tall” in Aus 0 height
  19. 19. Fuzzy Partition• Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:
  20. 20. Introduction toFuzzy Set Theory Set-Theoretic Operations
  21. 21. Set-TheoreticOperations• Subset A ⊆B ⇔µA ( x ) ≤ µ ( x ), ∀ ∈ B x U• Complement A = U − A ⇔ µA ( x ) = 1 − µA ( x )• Union C = A ∪ B ⇔ µC ( x ) = max( µA ( x ), µB ( x )) = µA ( x ) ∨ µB ( x ) Intersection• C = A ∩ B ⇔ µ ( x) = min( µ C A ( x ), µB ( x )) = µA ( x) ∧ µB ( x)
  22. 22. Set-TheoreticOperations A⊂ B A A∩ B A∪ B
  23. 23. PropertiesInvolution A=A De Morgan’s laws A∪ B = B ∪ A A∪ B = A∩ BCommutativity A∩ B = B ∩ A A∩ B = A∪ B ( A ∪ B) ∪ C = A ∪ ( B ∪ C )Associativity ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C )Distributivity A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) A∪ A = AIdempotence A∩ A = A A ∪ ( A ∩ B) = AAbsorption A ∩ ( A ∪ B) = A
  24. 24. Properties• The following properties are invalid for fuzzy sets: – The laws of contradiction A∩ A = ∅ – The laws of excluded middle A∪ A =U
  25. 25. Other Definitions for SetOperations• Union µ A∪ B ( x) = min ( 1, µ A ( x) + µ B ( x) )• Intersection µ A∩ B ( x) = µ A ( x) ×µ B ( x)
  26. 26. Other Definitions for SetOperations• Unionµ A∪ B ( x ) = µ A ( x ) + µ B ( x ) − µ A ( x ) µ B ( x )• Intersection µ A∩ B ( x) = µ A ( x) ×µ B ( x)
  27. 27. GeneralizedUnion/Intersection• Generalized Intersection t-norm• Generalized Union t-conorm
  28. 28. T-Norm Or called triangular norm. T :[0,1] × [0,1] → [0,1]1. Symmetry T ( x, y ) = T ( y , x )2. Associativity T (T ( x, y ), z ) = T ( x, T ( y, z ))3. Monotonicity x1 ≤ x2 , y1 ≤ y2 ⇒ T ( x1 , y1 ) ≤ T ( x2 , y2 )4. Border Condition T ( x,1) = x
  29. 29. T-Conorm Or called s-norm. S :[0,1] × [0,1] → [0,1]1. Symmetry S ( x, y ) = S ( y , x )2. Associativity S ( S ( x, y ), z ) = S ( x, S ( y, z ))3. Monotonicity x1 ≤ x2 , y1 ≤ y2 ⇒ S ( x1 , y1 ) ≤ S ( x2 , y2 )4. Border Condition S ( x, 0) = x
  30. 30. Fuzzy Relations Review Fuzzy Relations
  31. 31. R ⊆ A×BBinary Relation (R) b1 a1 b2 A a2 a3 b3 B b4 a4 b5 1 0 1 0 0 a1 Rb1 a1 Rb3 a2 Rb5 0 1 0 0 0 ( a1 , b1 ), ( a1 , b3 ), ( a2 , b5 ) MR =  R =  1 0 0 1 0 ( a3 , b1 ), ( a3 , b4 ), ( a4 , b2 )    0 1 0 0 0 a3 Rb1 a3 Rb4 a4 Rb2
  32. 32. The Real-Life Relation• x is close to y – x and y are numbers• x depends on y – x and y are events• x and y look alike – x and y are persons or objects• If x is large, then y is small – x is an observed reading and y is a corresponding action
  33. 33. Fuzzy RelationsA fuzzy relation R is a 2D MF: R = { ( ( x, y ), µ R ( x, y ) ) | ( x, y ) ∈ X × Y }
  34. 34. R = { ( ( x, y ), µ R ( x, y ) ) | ( x, y ) ∈ X × Y } Example (Approximate Equal)X = Y = U = {1, 2,3, 4,5}  1 0.8 0.3 0 0  1 u−v = 0 0.8 1 0.8 0.3 0     0.8 u − v = 1 M R =  0.3 0.8 1 0.8 0.3µ R (u, v) =  0.3 u − v = 2   0 otherwise  0 0.3 0.8 1 0.8  0  0 0.3 0.8 1  
  35. 35. Max-Min CompositionX Y Z R: fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. R 。 S: the composition of R and S. A fuzzy relation defined on X an Z.µ RoS (x, z ) = max y min ( µ R ( x, y ), µS ( y, z ) ) = ∨ y ( µ R ( x, y ) ∧ µ S ( y , z ) )
  36. 36. µ S o R (x, y ) = max v min ( µ R ( x, v), µ S (v, y ) ) ExampleR a b c d S α β γ1 0.1 0.2 0.0 1.0 a 0.9 0.0 0.32 0.3 0.3 0.0 0.2 b 0.2 1.0 0.83 0.8 0.9 1.0 0.4 c 0.8 0.0 0.7 0.1 0.2 0.0 1.0 d 0.4 0.2 0.3 0.9 0.2 0.8 0.4 min0.1 0.2 0.0 0.4 max RoS α β γ 1 0.4 0.2 0.3 2 0.3 0.3 0.3 3 0.8 0.9 0.8
  37. 37. Max-min composition is not mathematically tractable, therefore other compositions such as max-product composition have been suggested.Max-Product CompositionX Y Z R: fuzzy relation defined on X and Y. S: fuzzy relation defined on Y and Z. R 。 S: the composition of R and S. A fuzzy relation defined on X an Z. µ RoS (x, z ) = max y ( µ R ( x, y ) µ S ( y, z ) )
  38. 38. Dimension ReductionProjection RRY =  R ↓ Y    RX =  R ↓ X   
  39. 39. Dimension ReductionProjection RRY =  R ↓ Y    RX =  R ↓ X    = ∫ max µ R ( x, y ) / y = ∫ maxµ R ( x, y ) / x Y x X yµ RY ( y ) = max µ R ( x, y ) µ RX ( x) = max µ R ( x, y) y x
  40. 40. Dimension ExpansionCylindrical ExtensionA : a fuzzy set in X.C(A) = [A↑X×Y] : cylindrical extension of A. C ( A) = ∫ µ A ( x ) | ( x, y ) µC ( A ) ( x , y ) = µ A ( x ) X ×Y
  41. 41. Types of Fuzzy Relations R ( x, x) = 1 for all x ∈ X• Reflexive – Irreflexive R ( x, x) ≠ 1 for some x ∈ X – Antireflexive R ( x, x) ≠ 1 for all x ∈ X – Epsilon Reflexive R ( x, x) ≥ ε for all x ∈ X• Symmetric R ( x, y ) = R ( y, x) for all x ∈ X – Asymmetric R ( x, y ) ≠ R( y, x) for some x ∈ X – Antisymmetric R( x, y ) > 0 and R( y, x) > 0 → x = y for all x, y ∈ X
  42. 42. Types of Fuzzy Relations• Transitive (max-min transitive) R ( x, z ) ≥ max min[ R ( x, y ), R ( y , z )] for all x,z ∈X y∈Y – Non-transitive: For some (x,z), the above do not satisfy. – Antitransitive: R ( x, z ) < max min[ R ( x, y ), R ( y , z )] for all x,z ∈X y∈Y• Example: X = Set of cities, R=“very far” Reflexive, symmetric, non-transitive
  43. 43. Types of Fuzzy Relations• Transitive Closure – Crisp: Transitive relation that contains R(X,X) with fewest possible members – Fuzzy: Transitive relation that contains R(X,X) with smallest possible membership – Algorithm: 1. R = R ∪ R o R ). ( 2. If R ≠ R, make R = R and go to step 1 3. Stop : R = RT
  44. 44. Types of Fuzzy Relations• Fuzzy Equivalence or Similarity Relation – Reflexive, symmetric, and transitive – Decomposition: R=  ααR ⋅ α∈ ,1] [0 α R is a crisp equivalence relation. Set of partitions : ∏ ={π(αR ) | α∈ (R) [0,1]} – Partition Tree
  45. 45. Types of Fuzzy Relations• Fuzzy Compatibility or Tolerance Relation – Reflexive and symmetric – Maximal compatibility class and complete cover • Compatibility class Subset A of X such that < x, y >∈R • Maximal compatibility class: largest compatibility class • Complete cover: Set of maximal compatibility classes – Maximal alpha-compatibility class – Complete alpha-covers – Note: Relation from distance metrics forms tolerance relation in clustering.
  46. 46. Bibliography• J. R. Jang, C. Sun, E. Mizutani, “Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence, Prentice Hall• Slides and notes: http://equipe.nce.ufrj.br/adriano/fuzzy/bib

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