PART 8Approximate Reasoning

1. Fuzzy expert systems2. Fuzzy implications3. Selecting fuzzy implications4. Multiconditional reasoning5. Fuzzy relation equations6. Interval-valued reasoning

FUZZY SETS AND

FUZZY LOGICTheory and Applications

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Fuzzy expert systems

Fuzzy implications

Extensions of classical implications:

•

•

•

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1]. [0, allfor ) ),(() ,( a, bbacubaJ

on.intersectifuzzy continuous a denotes where

1], [0, allfor }) ,(|]1 ,0[sup{) ,(

i

a, bbxaixba J

laws.Morgen De esatisfy th torequired are where

), )),( ),((() ,(

)), ,( ),(() ,(

u, i, c

bbcaciuba

baiacuba

J

J

Fuzzy implications

S-implications

1.Kleene-Dienes implication

2.Reichenbach implication

3.Lukasiewicz implication

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1]. [0, allfor ) ,1max() ,( a, bbababJ

1]. [0, allfor 1) ,( a, bababarJ

1]. [0, allfor )1 ,1min() ,( a, bbabaaJ

Fuzzy implications

S-implications

4. Largest S-implication

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1]. [0, allfor

otherwise,

0when

1when

1

1) ,(

a, bb

a

a

b

baLSJ

Fuzzy implications

Theorem 8.1

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Fuzzy implications

R-implications

1. Gödel implication

2. Goguen implication

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.when

when

1}) ,min(|sup{) ,(

ba

ba

bbxaxbag

J

.when

when

/

1}|sup{) ,(

ba

ba

abbaxxba

J

Fuzzy implications

R-implications

3. Lukasiewicz implication

4. the limit of all R-implications

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).1 ,1min(

})1 ,0min(|sup{) ,(

ba

bxaxbaa

J

otherwise.

1when

1) ,(

ab

baLRJ

Fuzzy implications

Theorem 8.2

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Fuzzy implications

QL-implications

1. Zadeh implication

2. When i is the algebraic product and u is the algebraic sum.

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)].(min1[max) ,( a, ba, bam J

.1) ,( 2baabap J

Fuzzy implications

QL-implications

3. When i is the bounded difference and u is the bounded sum, we obtain the Kleene-Dienes implication.

4. When i = imin and u = umax

11., ba

, ba

a

a

b

baq

11when

11when

1when

1

1) ,(

J

Fuzzy implications

Combined ones

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Fuzzy implications

Axioms of fuzzy implications

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Fuzzy implications

Axioms of fuzzy implications

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Fuzzy implications

Axioms of fuzzy implications

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Fuzzy implications

Theorem 8.3

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Selecting fuzzy implications

Generalized modus ponens

any fuzzy implication suitable for approximate reasoning based on the generalized modus ponens should satisfy (8.13) for arbitrary fuzzy sets A and B.

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Selecting fuzzy implications

Theorem 8.4

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Selecting fuzzy implications

Theorem 8.5

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Selecting fuzzy implications

Generalized modus tollens

Generalized hypothetical syllogism

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Multiconditional reasoning

• general schema of multiconditional approximate reasoning

The method of interpolation is most common way to determine B‘. It consists of the following two steps:

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Multiconditional reasoning

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Multiconditional reasoning

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Multiconditional reasoning

four possible ways of calculating the conclusion B':

Theorem 8.624

Fuzzy relation equations

• Suppose now that both modus ponens and modus tollens are required. The problem of determining R becomes the problem of solving the following system of fuzzy relation equation:

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Fuzzy relation equations

Theorem 8.7

Fuzzy relation equations

If

then is also the greatest approximate solution to the system (8.30).

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Fuzzy relation equations

Theorem 8.8

Interval-valued reasoning

Let A denote an interval-valued fuzzy set.

LA,UA are fuzzy sets called the lower bound and the upper bound of A.

A shorthand notation of A( x )

When LA = UA, A becomes an ordinary fuzzy set.29

Interval-valued reasoning

given a conditional fuzzy proposition (if - then rule)

where A, B are interval-valued fuzzy sets defined on the universal sets X and Y.

given a fact

how can we derive a conclusion in the form

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Interval-valued reasoning

view this conditional proposition as an interval-valued fuzzy relation R = [LR,UR], where

It is easy to prove that LR (x, y) ≦ UR (x, y) and, hence, R is well defined.

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Interval-valued reasoning

Once relation R is determined, it facilitates the reasoning process. Given A’ = [LA’,UA’], we derive a conclusion B’ = [LB’,UB’] by the compositional rule of inference

where i is a t-norm and

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Interval-valued reasoning

Examplelet a proposition be given, where

Assuming that the Lukasiewicz implication

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Interval-valued reasoning

Exercise 8

• 8.2

• 8.4

• 8.8

• 8.9

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