[4] Fuzzy Logic and
Approximate Reasoning - 1
Teknik Informatika
Fuzzy Implication Rules
Reasoning Î generation of inferences from a given set of facts and rules
IF x is A1 THEN y is B1
IF x is A2 THEN y is B2
IF x is A3 THEN y is B3
Let x be a linguistic variable, and A1, A2, and A3 are three fuzzy sets Let y be a linguistic variable, and B1, B2, and B3 are three fuzzy sets Then the implication rules between
Fuzzy Implication Rules
IF x is Ai THEN y is Bi ;
Fuzzy Implication relations :
)]
,
(
),
,
[(
)
,
(
x
y
x
y
x
y
R
=
μ
RiFuzzy Implication Relations :
- Dienes – Rescher Implication - Lukasiewicz implication
Fuzzy Implication Rules
Dienes – Rescher Implication
)
Lukasiewicz Implication
)]
Æ replacing negation by one’s complement; and logical OR by sum (+) operator
Fuzzy Implication Rules
Mamdani Implication
)]
Zadeh Implication
)]
Æ representing logical AND by min, logical OR by max, and
negation by one’s complement
Fuzzy Implication Rules
Godel Implication
Fuzzy Logic
Typical Proportional Inference Rules
p q
p
q ∧ → ⇒ ¬
¬ ( )
q q
p
p ∧ ( → ) ⇒
Let p, q, and r be three propositions. Three proportional inference :
1. Modus Ponens , Given a proposition p and a propotional
implication rule pÆq, we can derive the inference q
r
p →
⇒ →
∧
→ q) (q r)
(p
2. Modus Tollens, Given a proposition ~ p and a propotional
implication rule pÆq, we can derive the inference ~ p
Fuzzy Logic
Fuzzy Extension of the Inference Rules
y is B’ I nferred :
x is A’ Given :
I F x is A THEN y is B Given :
Generalized Modus Ponens (GMP)
Production Rule : IF x is A THEN y is B
Fuzzy fact : x is A’
Fuzzy Logic
Computing Fuzzy Inference in GMP
Production Rule : IF x is A THEN y is B
Fuzzy fact : x is A’
Inference using GMP : y is B’
)
,
(
)
(
)
(
''
y
Ax
Rx
y
B
μ
μ
μ
=
o
) (
' x
A
μ
μ
R(x, y)) (
' y
B
μ
?
;
Fuzzy Logic
Example :
Fuzzy Logic
Fuzzy Extension of the Inference Rules
x is A’ I nferred :
y is B’ Given :
I F x is A THEN y is B Given :
Generalized Modus Tollens (GMT)
Production Rule : IF x is A THEN y is B
Fuzzy fact : y is B’
Fuzzy Logic
Computing Fuzzy Inference in GMT
Production Rule : IF x is A THEN y is B
Fuzzy fact : y is B’
Inference using GMP : x is A’
T R
B
A'(y)
μ
'(x) [μ
(x, y)]μ
= o) (
' x
B
μ
μ
R(x, y)) (
' y
A
μ
?
;
Fuzzy Logic
Fuzzy Extension of the Inference Rules
z is C’ I nferred :
I F y is B’ THEN z is C Given :
I F x is A THEN y is B Given :
Generalized Hypothetical Syllogism (GHS)
Production Rule : IF x is A THEN y is B, IF y is B’ THEN z is C. A, B, C are fuzzy sets, and B’ is close to B
