1. Ch.3 Fuzzy Rules and Fuzzy Reasoning
Extension principle,
Fuzzy relations,
Fuzzy if-then rules,
Compositional rule of inference,
Fuzzy reasoning

2. Extension Principle Generalizes crisp math concept to fuzzy framework.
Extends point to point mapping to mapping of FS’s.
Maps points (x1, x2, …,xn) in X to a crisp set Y to be generalized such that it maps fuzzy subsets of X to Y.
f : X Y

3. If more than one point is mapped to a point in Y,
Extension Principle
If more than one point is mapped to a point in Y,
Ex) A = { (-1, 0.5), (0, 0.8), (1, 1), (2, 0.4) }, and y = x2
Find B=[(y, μB(y))].
B = 0.8/0 + 1/ /4 x
μA(x)
y = x2
μB(y) -1
0.5 1
max(0.5, 1)=1
0.8 2
0.4 4

4. Extension Principle: Extension Principle
Let A1, A2, …, An be FS’s in X1, X2, …, Xn, respectively.
X = and f: X Y, y = f(x1, x2, …, xn), then
extension principle allows us to map FS’s in X to Y as follows:
B = f(A)
where B is the fuzzy image (set) in Y:

5. Ex) f maps ordered pairs from X1 and X2 to Y, where X1 = {-1, 0, 1},
Extension Principle
Ex) f maps ordered pairs from X1 and X2 to Y, where X1 = {-1, 0, 1},
X2 = {-2, 2 }, Y = {-2, -1, 2, 3} and y = f(x1, x2) = x12+x2.
FS A1 = 0.5/ / /1, A2 = 0.4/-2 + 1/2, then B = f(A1, A2) ?
μB(y = -1) = max { min[μA1(x1 = -1), μA2(x2 = -2)], min[μA1(x1 = 1), μA2(x2 = -2)] }
= 0.4
Similarly, μB(y = 3) = max[ 0.5, 0.9 ] = 0.9
Thus, B = 0.1/ / / /3 x1
μA1(x1) x2
μA1xA2(x1, x2)
y = x12+x2
μB(y) -1
0.5 -2
0.4 2 1 3
0.9
0.1

6. Extension Principle
Ex)

7. Extension Principle
Ex)

8. Ex) Extension Principle = bell(x; 1.5, 2, 0.5) if X Y y = f(x)-2 2 -3 -1 1 3 Y
y = f(x)
0.2
0.4
0.6
0.8
Membership Grades A
0.5 B
= bell(x; 1.5, 2, 0.5) if

9. Fuzzy Relations A fuzzy relation R is a 2D MF: Ex)
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).

10. Fuzzy Relations
The max-min composition of two fuzzy relations R1 (defined on X and Y) and R2 (defined on Y and Z):
Properties:
Associativity:
Distributivity over union:
Weak distributivity over intersection:
Monotonicity:

11. Max-product composition
Fuzzy Relations
Max-product composition
Max-* composition
where * is a T-norm operator.

12. Fuzzy Relations
Ex)
Max-min composition:
Max-product composition:
Derive the degree of relevance between 2 in X and a in Z based on R1 and R2
= “ x is relevant to y,”
= “ y is relevant to z,”
where

13. Fuzzy if-than rules
Linguistic variable: a variable whose values are words or sentences
in a natural language.
Ex) Speed is a linguistic variable
Age is a linguistic variable
Defn: Fuzzy variables are characterized by triples:
x = Name of the fuzzy variable
X = Universe of discourse
R(x) = Fuzzy restriction
Ex) x = fast,
R(x) = fuzzy set of fast
2. x = old,
R(x) = 0.1/ /30 + ··· + 1/70 + 1/80

14. Defn: A linguistic variable is characterized by a quintuple:
Fuzzy if-than rules
Defn: A linguistic variable is characterized by a quintuple:
where
x = Name of the linguistic variable
T(x) = A term set of x, i.e., the set of names of linguistic variables of x
with each value being a fuzzy variable defined in X
G = Syntactic rule for generating the names of the value of x
M = Semantic rule for associating each value of x with its meaning

15. Linguistic Hedge (Modifier)
With the linguistic hedges, we can define

16. Membership functions of the term set T(age)
Fuzzy if-than rules
Ex) Linguistic variable: Age, X = [0, 100]
T(age) = {young, not young, very young, …, middle aged,
not middle aged, …, old, not old, very old, not very old, …, not very young and not very old, … } 10 20 30 40 50 60 70 80 90
100
0.2
0.4
0.6
0.8 1
X = age
Membership Grades
Old
Very
Young
Middle Aged
Membership functions of the term set T(age)

17. Primary terms (young, middle aged, old)
Fuzzy if-than rules
Primary terms (young, middle aged, old)
Hedges (very, more or less, quite, extremely, etc)
Connectives (and, or, either, neither)
Operations on linguistic values:
Concentration (very) and dilation (more or less)
Contrast intensification
Concentration
Dilation

18. Ex) Fuzzy if-than rules NOT(A) = A AND B = A OR B =
more or less = DIL(old)
not young and not old

19. extremely old = CON(CON(CON(old))
Fuzzy if-than rules
young but not too young
extremely old = CON(CON(CON(old)) 10 20 30 40 50 60 70 80 90
100
0.2
0.4
0.6
0.8 1
X = age
Membership Grades
(a) Primary Linguistic Values
Old
Young
(b) Composite Linguistic Values
Not Young and Not Old
More or Less Old
Extremely Old
Young but
Not Too Young

20. Ex) Fuzzy if-than rules Solid line : A Dotted line : INT(A)1 2 3 4 5 6 7 8 9 10
0.2
0.4
0.6
0.8 X
Membership Grades
Effects of Contrast Intensifier
Solid line : A
Dotted line : INT(A)
Dashed line : INT(INT(A))
Dashed-dot line : INT(INT(INT(A)))

21. Orthogonality: Fuzzy if-than rules
A term set of a linguistic variable x on X is orthogonal if
where the ’s are convex and normal fuzzy sets.

22. Fuzzy if-than rules
Fuzzy if-then rules (fuzzy rule, fuzzy implication, fuzzy conditional statement) assumes the form:
If x is A, then y is B.
(antecedent, premise), (consequence, conclusion)
Two ways to interpret a fuzzy rule:
“If pressure is high, then volume is small”
A coupled with B: (A and B)
A entails B: (Not A or B) Y X A B {
A entails B
A coupled with B

23. Fuzzy relation: R = A B Fuzzy if-than rules
A coupled with B: (A and B)
A entails B: (Not A or B)
Material implication:
Propositional calculus:
Extended propositional calculus:
Generalization of modus ponens:
where is a T-norm operator
where and is a T-norm operator.

24. Fuzzy implication function f:
Fuzzy if-than rules
Fuzzy implication function f:
Transforming the membership grades of x in A and y in B into those of (x, y) in A B
A coupled with B
Min
Algebraic-product
Bounded product
Drastic product
A entails B
Zadeh’s arithmetic rule
Zadeh’s max-min rule
Boolean fuzzy implication
Goguen’s fuzzy implication

25. 1st row: 2nd row: Fuzzy if-than rules fuzzy implication ftns
corresponding R

26. Fuzzy if-than rules

27. Fuzzy Reasoning Logic Logic is a basis for reasoning.
How to combine two propositions:

28. Reasoning/Inference Rules
Modus Ponens (MP)
Modus Tollens (MT)
Hypothetical Syllogism (HS) 3단 논법
Ex: (1) Gildong is bald.
(2) Bald men are rich.
Gildong is rich.

29. p: This tomato is very red.
Generalized MP (GMP)
Tuned a little bit.
p: This tomato is very red.
Implication: If a tomato is red, then the tomato is ripe.
Conclusion: This tomato is very ripe.

30. n-valued Logic - 2-valued logic is not enough. Logic operations:
indeterminant.
Logic operations:
- n-valued logic:
Note: For

31. crisp (Koo is a student of EE).
Fuzzy Logic
Proposition:
Subject
crisp (Koo is a student of EE).
 Fuzzy Logic:
Proposition → Fuzzy proposition
Predicate → Fuzzy predicate
Truth value → A point in the unit interval [0,1]

32. Fuzzy Logic
(Gildong is young.) is very true or more or less true (one point in [0,1]).

33. Connections for truth value:
Fuzzy Logic
Defn: The truth value of the proposition “ X is A ” or simply truth
value of A is denoted by which is defined to be a point in [0,1]
(called numerical truth value) or a possible fuzzy set in [0,1] (called
linguistic truth value, e.g., very true)
Connections for truth value:

34. Fuzzy Logic
Defn:

35. Fuzzy Logic
Ex)

36. Fuzzy (or Approximate) Reasoning
Fuzzy proposition - conclusion
Fuzzy proposition
Rules of inference govern the deduction of a proposition
from a set of premises.
For fuzzy reasoning or approximate reasoning,
q = approximate rather than an exact consequence of

37. 4 Types of Rules for Approximate Reasoning
Categorical reasoning
Qualitative reasoning
Syllogistic reasoning
Dispositional reasoning
(1) Categorical reasoning
Premise: X is A
Implication: If X is C, then Y is B.
where X, Y, Z taking values in U, V, W, and
A, B, C are fuzzy predicates.

38. Approximate Reasoning
(a) Projection rules of inference

39. Approximate Reasoning
(b) Conjunction/Particularization
Y is “don’t care”

40. Approximate Reasoning
(c) Disjunction/Cartesian product
(d) Negation

41. Approximate Reasoning
(e) Entailment rule of inference
X is A
A B
X is B
Ex) Mary is very young
Very young young
Mary is young
(f) Compositional rule of inference
(X,Y) is R
Y is A R R

42. Approximate Reasoning
Ex) p = X is small
q = X and Y are approximately equal
U = {1, 2, 3, 4}
small = 1/ / /3
approxi. equal = 1/(1,1)+ 1/(2,2)+ 1/(3,3)+ 1/(4,4)+ 0.5/(1,2)+ 0.5/(2,1) /(2,3) +0.5/(3,2)+ 0.5/(3,4)+ 0.5/(4,3)
X is small (X,Y) approximately equal =

43. Approximate Reasoning
A possible interpretation of the inference may be the following:
X is small.
X and Y are approximately equal.
Y is more or less small.

44. Approximate Reasoning
(g) Generalized Modus Ponens (GMP)
X is A
If X is B, then Y is C
Y is
where
MP:
Partial matching by GMP.
(h) Extension principle
f(X) is f(A)
Ex) X is small
is small

45. Approximate Reasoning
(2) Qualitative Reasoning
I/O relation of a system is expressed as a collection of if-then rules in which the antecedents and consequents involve linguistic variables:
If X is and Y is , then Z is
If X is and Y is , then Z is
Ex) R1: If pressure is high, then volume is small.
R2: If pressure is low, then volume is large.
If pressure is medium, then volume is
where

46. Fuzzy reasoning (approximate reasoning):
An inference procedure that derives conclusion from a set of fuzzy rules and known facts
Compositional rule of inference y
y=f(x) b a x
a and b : points
y = f(x) : a curve
a and b : intervals
y = f(x) : an interval-valued function

47. Compositional rule of inference:
Fuzzy Reasoning
Compositional rule of inference:
Ex)
Let F : A B, fuzzy relation, and A: a fuzzy set of X.
Find the resulting fuzzy set B:
where denotes the composition operator.

48. Fuzzy Reasoning

49. Fuzzy reasoning (Generalized Modus Ponens)
Rule : “if the tomato is red, then it is ripe”
Fact : “the tomato is more or less red”
Infer : “the tomato is more or less ripe”
fact : x is A,
rule : If x is A, then y is B,
conclusion : y is B.
fact : x is A’,
rule : If x is A, then y is B,
conclusion : y is B’.
where A, B, A’ and B’ are fuzzy sets, and
A’ and B’ are close to A and B, respectively.

50. Def: Approximate reasoning (fuzzy reasoning)
Assume the fuzzy implication A B is expressed as a fuzzy
relation R on X x Y. Then the fuzzy set induced by “x is ”
and the fuzzy rule “If x is A, then y is B” is defined by
Or, equivalently,

51. Single rule with single antecedent
Fuzzy Reasoning
Single rule with single antecedent
Fact : x is A’
Rule : If x is A, then y is B
Conclusion : y is B’ A X w A’ B Y
x is A’ B’
y is B’
min

52. Single rule with multiple antecedents
Fuzzy Reasoning
Single rule with multiple antecedents
Fact : x is A’ and y is B’
Rule : If x is A and y is B, then z is C
Conclusion : z is C’

53. Thm 3.1: Decomposition method
Fuzzy Reasoning
Thm 3.1: Decomposition method
The resulting consequence can be expressed as the inferred fuzzy set of
a GMP problem for a single fuzzy rule with a single antecedent. A B
min X Y w A’ B’ C Z C’
x is A’
y is B’
z is C’ w1 w2

54. Multiple rules with multiple antecedents
Fuzzy Reasoning
Multiple rules with multiple antecedents
Fact : x is A’ and y is B’
Rule 1 : If x is A1 and y is B1, then z is C1
Rule 2 : If x is A2 and y is B2, then z is C2
Conclusion : z is C’

55. Graphic representation
Fuzzy Reasoning
Graphic representation A1 B1 A2 B2
min X Y w1 w2 A’ B’ C1 C2 Z C’
x is A’
y is B’
z is C’

56. Fuzzy reasoning divided into four steps:
Degrees of compatibility (match)
Compare the known facts with the antecedents of fuzzy rules
Firing strength
Degree to which the antecedent part of the rule is satisfied
Qualified (induced) consequent MFs
Apply the firing strength to the consequent MF of a rule
Overall output MF
Aggregate all the qualified consequent MFs