Vagueness

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 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]. mA: U [0, 1] mAA 1 young 0.5 0 Age 25 30 40

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] 1 0.5 0 Age 25 30 40

6. Fuzzy Set Representation • a-cuts: Aa= {u | A(u) a} Aa+= {u | A(u) >a}strong a-cut A0.5 = [0, 30] 1 0.5 0 Age 25 30 40

7. Fuzzy Set Representation • a-cuts: Aa= {u | A(u) a} Aa+= {u | A(u) >a}strong a-cut A(u) = sup {a| u Aa} A0.5 = [0, 30] 1 0.5 0 Age 25 30 40

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 definitionof 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:

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

15. Fuzzy Numbers • A fuzzy numberAis a fuzzy set onR: A must be a normal fuzzy set Aa must be a closed interval for every a(0, 1] supp(A)= A0+ must be bounded

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

17. Basic Types of Fuzzy Numbers 1 1 0 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: about 2 about 3 more or less 6 = about 2.about 3 1 0 + 2 3 6

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

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. Operations of Fuzzy Numbers • Interval-based operations: (A  B)a= Aa Ba

25. Possibility Theory • Possibility vs. Probability • PossibilityandNecessity

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

27. Possibility Theory • Membership degree = possibility degree

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

29. Possibility Theory • Derived properties: Nec() = 1 Nec() = 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. Possibility Theory

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, ...}

34. Multivalued Logic • Truth valuesare 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. Fuzzy Logic • ifx is Atheny 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 theny 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 Controller IF the temperature is very high AND the pressure is slightly low THEN the heat change should be sligthly negative

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

40. Fuzzification 1 0 x0

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.