REFERENSI KUMPULAN FUZZY | User Club Subang Fuzzy1

[4] Fuzzy Logic and
Approximate Reasoning - 1

Teknik Informatika
Universitas Trunojoyo Madura

Fuzzy Implication Rules
Reasoning Î generation of inferences from
a given set of facts and rules
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
variable x and y may be described as :

IF x is A1 THEN y is B1
IF x is A2 THEN y is B2
IF x is A3 THEN y is B3

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Fuzzy Implication Rules
IF x is Ai THEN y is Bi ;
Fuzzy Implication relations :

R( x, y ) = [( x, y ), μ Ri ( x, y )]

Fuzzy Implication Relations :
- Dienes – Rescher Implication
- Lukasiewicz implication
- Mamdani Implication
- Zadeh Implication
- Godel Implication

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Fuzzy Implication Rules
Dienes – Rescher Implication

p → q ≡ ¬p ∨ q

IF x is Ai THEN y is Bi ;

μ Ri ( x, y ) = Max[1 − μ Ai ( x), μ Bi ( x)]

Lukasiewicz Implication

p → q ≡ ¬p ∨ q
1 − μ Ai ( x) + μ Bi ( x)

Æ replacing negation by one’s complement;
and logical OR by sum (+) operator

Æ μ Ri ( x, y ) = min[1,1 − μ Ai ( x) + μ Bi ( x)]
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Fuzzy Implication Rules
Mamdani Implication
IF x is Ai THEN y is Bi ;

μ Ri ( x, y ) = Min[ μ Ai ( x), μ Bi ( x)]
μ Ri ( x, y ) = μ Ai ( x) μ Bi ( y )

Zadeh Implication
p → q ≡ ( p ∧ q ) ∨ (¬ p )

Æ representing logical AND by min,
logical OR by max, and
negation by one’s complement

μ Ri ( x, y ) = max[min(μ Ai ( x), μ Bi ( x)),1 − μ Ai ( x)]
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Fuzzy Implication Rules

Godel Implication
IF x is Ai THEN y is Bi ;

μ Ri ( x, y ) = 1 if μ Ai ( x) ≤ μ Bi ( x)
= μ Bi otherwise

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Fuzzy Logic
Typical Proportional Inference Rules
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

p ∧ ( p → q) ⇒ q

2. Modus Tollens, Given a proposition ~ p and a propotional

implication rule pÆq, we can derive the inference ~ p

¬q ∧ ( p → q ) ⇒ ¬p

3. Hypothetical Syllogism, Given 2 implication rule pÆq and qÆr,
we can derive implication pÆr

(p → q) ∧ (q → r) ⇒ p → r
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Fuzzy Logic
Fuzzy Extension of the Inference Rules
Generalized Modus Ponens (GMP)
Given :

I F x is A THEN y is B

Given :

x is A’

I nferred :

y is B’

Production Rule : IF x is A THEN y is B
Fuzzy fact
: x is A’
A, B, A’, B’ are fuzzy set such that A’ is close to A,
and B’ is close to B
Inference Rule : the closer the A’ to A, the closer B’ to B
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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’
μ B' ( y)

?

μ A' ( x)

;

μ R ( x, y )

μ B ' ( y ) = μ A' ( x) o μ R ( x, y )

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Æ max –min operator

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Fuzzy Logic
Example :
Compute membership of the inference generated using GMP.
Let
μ A' ( x) = [0.8 0.9 0.2]

⎡ 0.8 0.6 0.5⎤
⎢
⎥
μ R ( x, y ) = ⎢0.6 0.5 0.9⎥
⎢⎣0.7 0.6 0.5⎥⎦

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Fuzzy Logic
Fuzzy Extension of the Inference Rules
Generalized Modus Tollens (GMT)

Given :

I F x is A THEN y is B

Given :

y is B’

I nferred :

x is A’

Production Rule : IF x is A THEN y is B
Fuzzy fact
: y is B’
Inference Rule : the more is difference between B’ and B,
the more difference between A’ and A
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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’
μ A' ( y )

?

μ B ' ( x)

;

μ R ( x, y )

μ A' ( y ) = μ B ' ( x) o [ μ R ( x, y )]T

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Æ max –min operator

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Fuzzy Logic
Fuzzy Extension of the Inference Rules
Generalized Hypothetical Syllogism (GHS)
Given :

I F x is A THEN y is B

Given :

I F y is B’ THEN z is C

I nferred :

z is C’

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
Inference Rule : the closer B’ to B, the closer C’ to C

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