1. 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. 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. 3 Fuzzy Sets Zadeh, L.A. (1965). Fuzzy Sets Journal of Information and Control

4. 4 Fuzzy Set Definition A 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 2540 Age 30 0.5 young

5. 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 2540 Age 30 0.5

6. 6 Fuzzy Set Representation  -cuts: A  = {u | A(u)   } A  = {u | A(u)   } strong  -cut A 0.5 = [0, 30] 0 1 2540 Age 30 0.5

7. 7 Fuzzy Set Representation  -cuts: A  = {u | A(u)   } A  = {u | A(u)   } strong  -cut A(u) = sup {  | u  A  } A 0.5 = [0, 30] 0 1 2540 Age 30 0.5

8. 8 Fuzzy Set Representation Support: supp(A) = {u | A(u) > 0} = A 0+ Core: core(A) = {u | A(u) = 1} = A 1 Height: h(A) = sup U A(u)

9. 9 Fuzzy Set Representation Normal fuzzy set: h(A) = 1 Sub-normal fuzzy set: h(A)  1

10. 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. 11 Membership Degrees Voting model: P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 P8P8 P9P9 P 10 1 2 3 xx 4 xxxxx 5 xxxxxxxxx 6 xxxxxxxxxx

12. 12 Fuzzy Subset Relations A  B iff A(u)  B(u) for every u  U A is more specific than B “X is A” entails “X is B”

13. 13 Fuzzy Set Operations Standard definitions: Complement: A(u) = 1  A(u) Intersection: (A  B)(u) = min[A(u), B(u)] Union: (A  B)(u) = max[A(u), B(u)]

14. 14 Fuzzy Set Operations Example: not young = young not old = old middle-age = not young  not old old =  young

15. 15 Fuzzy Numbers A fuzzy number A is a fuzzy set on R: A must be a normal fuzzy set A  must be a closed interval for every  (0, 1] supp(A) = A 0+ must be bounded

16. 16 Basic Types of Fuzzy Numbers 1 0 1 0 1 0 1 0

17. 17 Basic Types of Fuzzy Numbers 1 0 1 0

18. 18 Operations of Fuzzy Numbers Extension principle for fuzzy sets: f: U 1 ...  U n  V induces g: U 1 ...  U n  V ~ ~ ~

19. 19 Operations of Fuzzy Numbers Extension principle for fuzzy sets: f: U 1 ...  U n  V induces g: U 1 ...  U n  V [g(A 1,...,A n )](v) = sup {(u1,...,un) | v = f(u1,...,un)} min{A 1 (u 1 ),...,A n (u n )} ~ ~ ~

20. 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. 21 Operations of Fuzzy Numbers EP-based operations: ++ 1 about 2more or less 6 = about 2.about 3about 3 0 236

22. 22 Operations of Fuzzy Numbers Arithmetic operations on intervals: [a, b]  [d, e] = {f  g | a  f  b, d  g  e}

23. 23 Operations of Fuzzy Numbers Arithmetic operations on intervals: [a, b]  [d, e] = {f  g | 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. 24 Operations of Fuzzy Numbers Interval-based operations: (A  B)   = A   B 

25. 25 Possibility Theory Possibility vs. Probability Possibility and Necessity

26. 26 Possibility Theory Zadeh, L.A. (1978). Fuzzy Sets as a Basis for a Theory of Possibility Journal of Fuzzy Sets and Systems

27. 27 Possibility Theory Membership degree = possibility degree

28. 28 Possibility Theory Axioms: 0  Pos(A)  1 Pos(  ) = 1Pos(  ) = 0 Pos(A  B) = max[Pos(A), Pos(B)] Nec(A) = 1 – Pos(A)

29. 29 Possibility Theory Derived properties: Nec(  ) = 1Nec(  ) = 0 Nec(A  B) = min[Nec(A), Nec(B)] max[Pos(A), Pos(A)] = 1 min[Nec(A), Nec(A)] = 0 Pos(A) + Pos(A)  1 Nec(A) + Nec(A)  1 Nec(A)  Pos(A)

30. 30 Possibility Theory ProbabilityPossibility p: U  [0, 1] Pro(A) =  u  A p(u) r: U  [0, 1] Pos(A) = max u  A r(u) Normalization:  u  U p(u) = 1max u  U r(u) = 1 Additivity: Pro(A  B) = Pro(A) + Pro(B)  Pro(A  B)Pos(A  B) = max[Pos(A), Pos(B)] Pro(A) + Pro(A) = 1Pos(A) + Pos(A)  1 Total ignorance: p(u) = 1/|U| for every u  Ur(u) = 1 for every u  U Probability-possibility consistency principle: Pro(A)  Pos(A)

31. 31 Fuzzy Relations Crisp relation: R(U 1,..., U n )  U 1 ...  U n R(u 1,..., u n ) = 1 iff (u 1,..., u n )  R or = 0 otherwise

32. 32 Fuzzy Relations Crisp relation: R(U 1,..., U n )  U 1 ...  U n R(u 1,..., u n ) = 1 iff (u 1,..., u n )  R or = 0 otherwise Fuzzy relation is a fuzzy set on U 1 ...  U n

33. 33 Fuzzy Relations Fuzzy relation: U 1 = {New York, Paris}, U 2 = {Beijing, New York, London} R = very far R = {(NY, Beijing): 1,...} NYParis Beijing1.9 NY0.7 London.6.3

34. 34 Multivalued Logic Truth values are in [0, 1] Lukasiewicz:  a = 1  a a  b = min(a, b) a  b = max(a, b) a  b = min(1, 1  a + b)

35. 35 Fuzzy Logic if x is A then y is BR(u, v)  A(u)  B(v) x is A’A’(u) ------------------------ ---------------------------------------- y is B’B’(v)

36. 36 Fuzzy Logic if x is A then y is B x is A’ ------------------------ y is B’ B’ = B +  (A/A’)

37. 37 Fuzzy Controller As special expert systems When difficult to construct mathematical models When acquired models are expensive to use

38. 38 Fuzzy Controller IF the temperature is very high AND the pressure is slightly low THEN the heat change should be sligthly negative

39. 39 Fuzzy Controller Controlled process Defuzzification model Fuzzification model Fuzzy inference engine Fuzzy rule base actions conditions FUZZY CONTROLLER

40. 40 Fuzzification 1 0 x0x0

41. 41 Defuzzification Center of Area: x = (  A(z).z)/  A(z)

42. 42 Defuzzification Center of Maxima: M = {z | A(z) = h(A)} x = (min M + max M)/2

43. 43 Defuzzification Mean of Maxima: M = {z | A(z) = h(A)} x =  z/|M|

44. 44 Exercises In Klir’s FSFL: 1.9, 1.10, 2.11, 4.5, 5.1 (a)-(b), 8.6, 12.1.