Fuzzy logic was introduced by Lotfi Zadeh in 1965 to address problems with classical logic being too precise. Fuzzy logic allows for truth values between 0 and 1 rather than binary true/false. It involves fuzzy sets, membership functions, linguistic variables, and fuzzy rules. Fuzzy logic can be applied to knowledge representation and inference using concepts like fuzzy predicates, relations, modifiers and quantifiers. It has various applications including household appliances, animation, industrial automation, and more.

1. Fuzzy logic
and application in AI
Innopolis University: ToC
Ildar Nurgaliev

2. History
Lotfi A. Zadeh
Introduced:
By Lotfi Zadeh in 1965
Problem:
Most classes are too precise.
We need to generalize that and
introduce a class whose
boundaries are un-sharp

3. History
"Fuzzy theory is wrong, wrong, and pernicious. What we need is more logical
thinking, not less. The danger of fuzzy logic is that it will encourage the sort of
imprecise thinking that has brought us so much trouble. Fuzzy logic is the
cocaine of science."
Professor William Kahan UC Berkeley
"’Fuzzification’ is a kind of scientific permissiveness."
Professor Rudolf Kalman UFlorida

4. History

5. Classical logic
Proposition - a sentence with a truth value.
Truth value - true (1 or T) or false (0 or F)

6. Classical logic
Logic operations:

7. Classical logic
Logic operations:

8. Classical logic
Logic formula:
● Each propositional variable is a formula
● If X is a formula, then ¬X is a formula
● If X and Y are formulas, then X * Y is a formula, where
* is any binary connective

9. Fuzzy logic
Truth value ranges between 0 (completely false)
and 1 ( completely true).

10. Fuzzy logic
Logic operations:
¬X = (1-truth(X))
X AND Y = minimum(truth(X), truth(Y))
X OR Y = maximum(truth(X), truth(Y))
X ⊃ Y = maximum(minimum(truth(X),truth(Y)),1-truth(X))

11. First order logic
-Fuzzy modifier
-Fuzzy Truth quantifier
-Linguistic variable
-Fuzzy predicate

12. Predicate
A “predicate” is a group of words like
applies to Objects
Socrates
Tree
Two
That hat
Predicates
is a man
is green
is less than
belongs to

13. Fuzzy Predicate
A fuzzy predicate is a predicate whose
definition contains ambiguity
“z is expensive.” “w is young.”

14. How to interpret Fuzzy Predicate?
-P(x) is a fuzzy set.
-evaluated by membership function μP(x)
“x is P” x - variable
P - ambiguity set

15. Membership function
Example:
Watson used these functions for reasoning degree of true
Conjunction rule: μA⋀B(x)=min[μA(x), μB(x)]
Disjunction rule: μA⋁B(x)=max[μA(x), μB(x)]
Negation rule: μㄱA(x)=1-μA(x)

16. Fuzzy Modifier
“w is young.”

17. Fuzzy Modifier
“w is young.”
add the modifier “very”

18. Fuzzy Modifier
“w is young.”
add the modifier “very”

19. Fuzzy Modifier
T(“Age”) = {young, very young, very very young, … }
μvery young(u) = (μyoung(u))2
“w is young.”
add the modifier “very”

20. Linguistic variable
Linguistic variable = (x, T(x), U, G, M)

21. Linguistic variable
Linguistic variable = (x, T(x), U, G, M)
x: name of variable

22. Linguistic variable
Linguistic variable = (x, T(x), U, G, M)
x: name of variable
T(x): set of linguistic terms which can be a value of the variable

23. Linguistic variable
Linguistic variable = (x, T(x), U, G, M)
x: name of variable
T(x): set of linguistic terms which can be a value of the variable
U: set of universe of discourse which defines the characteristics of the variable

24. Linguistic variable
Linguistic variable = (x, T(x), U, G, M)
x: name of variable
T(x): set of linguistic terms which can be a value of the variable
U: set of universe of discourse which defines the characteristics of the variable
G: syntactic grammar which produces terms in T(x)

25. Linguistic variable
Linguistic variable = (x, T(x), U, G, M)
x: name of variable
T(x): set of linguistic terms which can be a value of the variable
U: set of universe of discourse which defines the characteristics of the variable
G: syntactic grammar which produces terms in T(x)
M: semantic rules which map terms in T(x) to fuzzy sets in U

26. Linguistic variable

27. Linguistic variable
X={“Hot”, T(Hot), U,G(Hot), M(hot)}
name : “Hot”,
T(Hot) : (‘warm’, ‘hot’, ‘very hot’),
U : ( [0..100] ),
G(Hot) : { ‘warm’ } ∪ { ‘hot’ } ∪ Ti+1 = { ‘very’ . Ti } )
M(hot) = { (u, μhot(u)) | u •∊ U }
ps: Hot - linguistic variable
hot - predicate

28. Linguistic variable
Directions:
1. Empty contents into saucepan; add 4½ cups (1 L) cold
water.
2. Bring to a boil, stirring constantly.
3. Reduce heat; partially cover and simmer for 15
minutes, stirring occasionally.
4 to 6 servings, 4½ cups (1 L)
Example: Linguistic variables in soup instructions

29. Fuzzy Truth Values
The qualifiers in T define “fuzzy truth values” and they can be defined
by the μP(x) (membership functions).

30. Fuzzy Truth Values
fuzzy truth qualifier is defined in the universal set
U = {Q | Q ∊ [0,1]}.
The qualifiers in T define “fuzzy truth values” and they can be defined
by the μP(x) (membership functions).

31. Fuzzy Truth Values
fuzzy truth qualifier is defined in the universal set
U = {Q | Q ∊ [0,1]}.
T = {true, very true, fairly true, absolutely true, … , absolutely false, fairly false, false}
The qualifiers in T define “fuzzy truth values” and they can be defined
by the μP(x) (membership functions).

32. Truth qualifiers by μP(x)

33. LETS START PARTY!
The Application of Fuzzy logic in AI
-Knowledge Base (KB)

34. Inference and Knowledge
Representation
rule type: if-then

35. Inference and Knowledge
Representation
(1) Modus ponens
Fact: x is A
Rule: If (x is A) then (y is B)
Result: y is B
rule type: if-then

36. Inference and Knowledge
Representation
(1) Modus ponens
Fact: x is A
Rule: If (x is A) then (y is B)
Result: y is B
(2) Modus tollens
Fact: y is ㄱB
Rule: If (x is A) then (y is B)
Result: x is ㄱA
ps: The modus ponens is used in the forward inference and
the modus tollens is in the backward one.
rule type: if-then

37. Representation of Fuzzy Predicate
“x is P”, it is represented by:
- fuzzy set P(x)
- membership function μP(x)(x)

38. Representation of Fuzzy Predicate
“x is P”, it is represented by:
- fuzzy set P(x)
- membership function μP(x)(x)
Fuzzy Relation
R = { ( x, μR(x)) | μR(x) ⩾ 0, x •∊ A}

39. Representation of Fuzzy Predicate
“x is P”, it is represented by:
- fuzzy set P(x)
- membership function μP(x)(x)
Fuzzy Relation
R = { ( x, μR(x)) | μR(x) ⩾ 0, x •∊ A}
Represent a predicate by
fuzzy relation:
R(x) = P

40. Representation of Fuzzy Rule
If x is A, then y is B
or
If A(x), then B(y)

41. Representation of Fuzzy Rule
If x is A, then y is B
or
If A(x), then B(y)
R(x, y): If A(x), then B(y)
or
R(x, y): A(x)➝B(y)
R = { ( (x, y), μR(x, y)) | μR(x, y) ⩾ 0, x •∊ A, y ∊ B}

42. Inference
(1) Generalized modus ponens (GMP)
Fact: x is A : R(x)
Rule: If (x is A) then (y is B) : R(x, y)
Result: y is B : R(y) =
R(x) o R(x, y)

43. Inference Example
KB: (x is A)➝(y is B), x is A
I R(x,y) = A × B
II fact ‘x is A’ into the form R(x)
III R(y) = R(x) o R(x, y)
μR(y) = ⋁ [μR(x) š⋀ μR(x, y)]
x

44. Inference Example
KB: (x is A)➝(y is B), x is A

45. Fuzzy logic application
● household appliances
● animation systems
● industrial automation
● chemical industry
● aerospace
● robotics
● mining and metal
processing
● transportation

46. Thank you for attention
References:
1) ‘First Course on Fuzzy Theory and Applications’, Kwang H. Lee
2) ‘Linguistic Variables: Clear Thinking with Fuzzy Logic’, Walter Banks