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.
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
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
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))
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)
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
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).
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
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
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
Notes de l'éditeur
… Classical logic
proposition: a sentence having truth value true (1) or false (0)
truth value: proposition o {0, 1}
logic variable: variable representing a proposition
… Logic operation
negation (NOT):CP
conjunction (AND): a š b
disjunction (OR): a › b
implication (o): a o b
… Logic function
logic function
logic primitive
logic formula
Fuzzy logic
fuzzy logic formula
fuzzy proposition 214 8. Fuzzy Logic
truth value: fuzzy proposition o [0, 1]
… Fuzzy logic operation
negation (NOT):CP
conjunction (AND): a š b
disjunction (OR): a › b
implication (o): Min(1, 1 b a)
Tautology
tautology: logic formula whose value is always true
inference: developing new facts by using the tautology
… Deductive inference
modus ponens
modus tollens
hypothetical syllogism
… Predicate logic
predicate
predicate logic
predicate proposition: proposition consist of predicate and object
evaluation of proposition
… Quantifier
universal quantifier: (for all)
existential quantifier: (there exists)
… Fuzzy predicate
fuzzy predicate: predicate represented by fuzzy sets
fuzzy truth value [0, 1]
fuzzy modifier
… Fuzzy truth qualifier
fuzzy truth value: true, very true, fairly true, etc.
Pvery true(v) = (true(v))2
value of “P is very true” is 0.81 when value of P is 0.9
As we know now, a predicate proposition in the classical logic has the
following form.
“x is a man.”
“y is P.”
x and y are variables, and “man” and “P” are crisp sets. The sets of
individuals satisfying the predicates are written by “man(x)” and “P(y)”.
Definition (Fuzzy predicate) .A fuzzy predicate is a predicate whose
definition contains ambiguity Ƒ
Example 8.16 For example,
“z is expensive.”
“w is young.”
The terms “expensive” and “young” are fuzzy terms. Therefore the
sets “expensive(z)” and “young(w)” are fuzzy sets. Ƒ
When a fuzzy predicate “x is P” is given, we can interpret it in two
ways.
(1) P(x) is a fuzzy set. The membership degree of x in the set P is
defined by the membership function PP(x).
(2) PP(x) is the satisfactory degree of x for the property P. Therefore, the
truth value of the fuzzy predicate is defined by the membership
function.
Truth value = PP(x)
The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely defined sets, not likelihood of some event or condition.
ps: In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A.
Knowledge Base is a physical system so we have to represent Predicate to another form for presenting in KB.
For this reason we use Fuzzy Relation.
The fuzzy rule may include fuzzy predicates in the antecedent and
consequent,
The operation used in the
reasoning is denoted by the notation “o”, and thus the result is represented
by the output of the composition when we use the GMP.