REFERENSI KUMPULAN FUZZY | User Club Subang Fuzzy1

[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 A_i THEN y is B_i ;

Fuzzy Implication relations :

)]

,

(

),

,

[(

)

,

(

x

y

x

y

x

y

R

=

μ

_Ri

Fuzzy 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

A

x

R

x

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