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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
2
Fuzzy expert systems
Fuzzy implications
Extensions of classical implications:
•
•
•
3
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
4
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
5
1]. [0, allfor
otherwise,
0when
1when
1
1) ,(
a, bb
a
a
b
baLSJ
Fuzzy implications
Theorem 8.1
6
Fuzzy implications
R-implications
1. Gödel implication
2. Goguen implication
7
.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
8
).1 ,1min(
})1 ,0min(|sup{) ,(
ba
bxaxbaa
J
otherwise.
1when
1) ,(
ab
baLRJ
Fuzzy implications
Theorem 8.2
9
Fuzzy implications
QL-implications
1. Zadeh implication
2. When i is the algebraic product and u is the algebraic sum.
10
)].(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
13
Fuzzy implications
Axioms of fuzzy implications
14
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
19
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
22
Multiconditional reasoning
23
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:
25
Fuzzy relation equations
Theorem 8.7
Fuzzy relation equations
If
then is also the greatest approximate solution to the system (8.30).
27
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
30
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
35