EE-646

Lecture-9

Fuzzy Implications

Introduction

• A fuzzy rule generally assumes the form

R: IF x is A, THEN y is B. Where A and B are linguistic values defined by fuzzy sets on UoD X and Y, respectively.

• The rule is also called a “fuzzy implication” or fuzzy conditional statement.

• Fuzzy implication is an important connective in fuzzy control systems because the control strategies are embodied by sets of IF-THEN rules

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Contd...

• Sometimes, the fuzzy rule is abbreviated as R: A → B or simply A → B

• In essence, the expression describes a relation between two variables x and y.

• This suggests that a fuzzy rule can be defined as a binary relation R on the product space X × Y

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Different Interpretations

1. “A is coupled with B”

is T-norm operator

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( ) ( )

( , )

A B

X Y

x yR A B A B

x y

Interpretations

2. “A entails B”→ Four different formulae

a. Material Implication

b. Propositional calculus

c. Extended Propositional calculus

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R A B A B

R A B A A B

R A B A B B

Interpretations

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d. Generalized Modus Ponens (GMP)

All the above four formulae reduce to the familiar identity

( , ) sup | ( ) ( ),0 1R A Bx y c x c y c

A B A B

Contd...

• Based on these two interpretations & various T-norm/co-norm operators, a no. of qualified methods can be formulated to calculate the fuzzy relation R: A → B

• Relation R can be viewed as fuzzy set with 2D MF

• f is called fuzzy implication function & performs the task of transforming the membership grades of x in A & y in B into those of (x, y) in A → B

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( , ) ( ), ( ) ( , )R A Bx y f x y f a b

Contd...

• For the first interpretation, “A is coupled with B” 4 different fuzzy relations A → B result from employing the most commonly used T-norm operators (min, algebraic, bounded, drastic product)

• For the second interpretation, “A entails B”, again 4 different fuzzy relations A → B have been reported in literature (Zadeh’s arithmetic rule, Zadeh’s max-min rule, Boolean, Goguen’s)

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Different Implications

1. Kleene-Diene’s Implication

2. Lukasiewicz’s "

3. Zadeh’s "

4. Stochastic "

5. Goguen’s "

6. Gödel’s "

7. Sharp "

8. General "

9. Mamdani’s "

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1. Kleene Diene’s Implication

Where, C is the cylindrical extension

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K 'y yR C A C B

K

max 1 ( ), ( )R A Bx y

2. Lukasiewicz’s Implication

is bounded sum

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L 'y yR C A C B

L

, min 1,1 ( ) ( )R A Bx y x y

3. Zadeh’s Implication

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Z 'y y yR C A C B C A

Z

, max min ( ), ( ) , 1 ( )R A B Ax y x y x

4. Stochastic Implication

It is based on the following equality

Defined as

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1 ( ) ( ) ( )B

P P A P A P BA

St 'y y yR C A C A C B

st , min 1,1 ( ) ( ) ( )R A A Bx y x x y

5. Goguen’s Implication

One of the requirements in multi-valued logic is that A → B should satisfy

This goal is achieved when we use the definition

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( ) ( , ) ( )A A B Bx x y y

GN

GN min 1, ( ) ( )

( )min 1,

( )

y y

AR

B

R C A C B

x

y

6. Gödel’s Implication

It is one of the best known implication formulae in multi-valued logic. It is defined as:

e.g.

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1, ( ) ( )

( ),otherwiseg

A B

R

B

x y

y

(0.5,0.7) 1 and (0.8,0.6) 0.6g g

A B A B

6. Gödel’s Implication...contd

Above definition results in the following fuzzy relation that is frequently used in fuzzy logic

or,

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( ) ( )g y ygR C A C B

( , ) ( ) ( )gR A Bg

x y x y

7. Sharp Implication

Looks like Gödel’s but it is more restrictive. It is defined as:

e.g.

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1, ( ) ( )

0, ( ) ( )s

A B

A B

A B

x y

x y

(0.5,0.7) 1 and (0.8,0.6) 0s s

A B A B

7. Sharp Implication...contd

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( ) ( )s y ysR C A C B

( , ) ( ) ( )sR A Bs

x y x y

8. General Implication

A very general implication which does not have any explicit name in the literature may be considered as combination of Gödel and sharp. It is based on the following formula:

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A B A B A B

8. General Implication...contd

Where α and β may be s or g. In fuzzy terms, Rαβ is defined as:

This relation is hardly ever used in the literature.

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( ) ( ) ,

( , ) min

1 ( ) 1 ( )

A B

R

A B

x y

x y

x y

9. Mamdani’s Implication

W. r. t . Fuzzy control this is the most important (and most simplest) known in the literature. Its definition is based on the intersection operation

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M

M ( ) ( )

( , ) min ( ), ( )

y y

R A B

A B A B

R C A C B

x y x y

9. Mamdani...Remarks

i. Also known as control implication, it is better than the conventional PI controller.

ii. Majority of applications are through Mamdani Implication.

iii. Sometimes subscript M is also written as c (for conjunction)

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Problem Task

Suppose there is a rule “IF x is A THEN y is B”, where the meanings of x is A & y is B are given as:

Determine all the implications for these rule sets

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1 2 3 4

1 2 3

0.1 0.4 0.7 1

0.2 0.5 0.9

Ax x x x

By y y

Further Reading

1. Jang et. al., “Neuro-Fuzzy & Soft Computing”, PHI, 1997.

2. Driankov, D. et. al., “An Introduction to Fuzzy Control”, Narosa, 2001.

3. Mamdani, E. H., “Application of Fuzzy Algorithm for Control of Simple Dynamic System”, Proc. IEEE, 121(12), 1974.

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