Upload
vuduong
View
233
Download
3
Embed Size (px)
Citation preview
Models for Inexact Reasoning
Fuzzy Logic – Lesson 9Selection of Fuzzy Implications
Master in Computational Logic
Department of Artificial Intelligence
Fuzzy Implications
• A FI is a function J: [0, 1] × [0, 1]→ [0, 1]
– Input: truth values a, b of given fuzzy propositions p, q
– Output: truth value of the conditional proposition “if p, then q”
• In classical logic, J can be defined in several distinct forms
– All these forms are equivalent
– This does not happen in fuzzy logic
Classical Implications
• Different ways of defining implications in classical logic
( , )J a b a b= ∨
{ }{ }( , ) max 0,1 |J a b x a x b= ∈ ∧ ≤
( , ) ( )J a b a a b= ∨ ∧
( , ) ( )J a b a b b= ∧ ∨
• They can be easily proved to be equivalent
Families of FIs
• Fuzzy counterparts of classical implications
( , ) ( ( ), )J a b S C a b=
[ ]{ }( , ) sup 0,1 | ( , )J a b x T a x b= ∈ ≤
( , ) ( ( ), ( , ))J a b S C a T a b=
( , ) ( ( ( ), ( )), )J a b S T C a C b b=
• They are not equivalent
– In general, the law of absorption of negation does not hold in Fuzzy Logic
Families of FIs
• These four general implications yield distinct families or classes of FIs
• Variants for each family are obtained by combining different T, S and C operators
• Each class of fuzzy implication has different properties
• Some FLs may belong to more than one class
S-Implications• Based on J(a,b) = S(C(a),b) and the standard fuzzy
complement
• Differ from one another by the chosen S
Implication S J
Kleene-Dienes
Reichenbach
Lukasiewicz
Largest S-implication
( ) ( ), max ,S a b a b=
( , )S a b a b ab= + −
( ) ( ), min 1,S a b a b= +
( )0
, 0
1
a b
S a b b a
otherwise
=
= =
( ), max(1 , )bJ a b a b= −
( ), 1RJ a b a ab= − +
( ) ( ), min 1,1aJ a b a b= − +
1 0
( , ) 1
1
LS
a b
J a b b a
otherwise
− =
= =
• Ordering of S-implications: JLS ≥ Ja ≥ Jr ≥ Jb
R-Implications• Based on the standard fuzzy complement and different
T-norms and:
Implication T J
Gödel
Goguen
Lukasiewicz
Largest R-implicationN/A—Least upper bound.
Cannot be obtained using the formula
( ) ( ), min ,T a b a b=
( , )T a b ab=
( ) ( ), max 0, 1T a b a b= + −
( ) ( ){ }1
, sup | min ,g
a bJ a b x a x b
b a b
≤= ≤ =
>
( ) { }1
, sup |
a b
J a b x ax b ba b
a
∆
≤
= ≤ = >
( ) ( ){ }( )
, sup | max 0, 1
min 1,1
aJ a b x a x b
a b
= + − ≤ =
− +
1( , )
1LR
b aJ a b
otherwise
==
• Ordering of R-implications: JLR ≥ Ja ≥ JΔ ≥ Jg
[ ]{ }( , ) sup 0,1 | ( , )J a b x T a x b= ∈ ≤
QL-Implications (Quantum-Logic)
• Based on dual T-S operators w.r.t the standard C and:
( , ) ( ( ), ( , ))J a b S C a T a b=
Implication Dual T-S J
Zadeh
Klir and Yuan 1
Kleene-Dienes
Klir and Yuan 2
min, maxT S= =
,T ab S a b ab= = + −
( )
( )
max 0, 1
min 1,
T a b
S a b
= + −
= +
( ) ( )( ), max 1 , min ,mJ a b a a b= −
( ) 2, 1PJ a b a a b= − +
( ) ( ), max 1 ,bJ a b a b= −
1
( , ) 1 1, 1
1 1, 1q
b a
J a b a a b
a b
=
= − ≠ ≠ ≠ =
min
max
ii
ii
T T
S S
=
=
Other Implications
• The previous families are the most widely used
• Other FIs are also possible:
– Using S(T(C(a),C(b)),b) as a source
– Combining existing FIs
( )
( )
( ) [ ]
( )
( ) [ ]
, min ( , ), (1 ,1 )
, min ( , ), (1 ,1 )
, min ( , ), (1 ,1 )
, min ( , ), (1 ,1 )
, min ( , ), (1 ,1 )
sg s g
gs g s
ss s s
gg g g
J a b J a b J b a
J a b J a b J b a
J a b J a b J b a
J a b J a b J b a
J a b J a b J b a∆ ∆
= − −
= − −
= − −
= − −
= − −VV
Selection of Fuzzy Implications
• Recall from the generalized modus ponens that:
' 'T
B A R= o
• The classical MP assumes that A’ = A, and thus obtains the conclusion “Y is B”:
T
B A R= o
• Which FIs satisfy the previous equation?
– Thus supporting the classical MP
Selection of Fuzzy Implications
• Theorem 1: Given A, T and J being:
– A: a normal fuzzy set
• (normal = )
– T: a continuous T-norm
– J: J be a FI derived from T( ( ), ( )) ( ( ), ( ))A B T A BJ x x x xµ µ ω µ µ=
• Then, the classical modus ponens holds
( )( ) sup ( ), ( ( ), ( ))A A Bx X
B y T x J x yµ µ µ∈
=
0 0| ( ) 1Ax X xµ∃ ∈ =
Selection of Fuzzy Implications
• Theorem 2: If the range of μA covers the whole interval [0, 1], then the FIs:
( )
( )
( )( )
1Gaines-Rescher ,
0
1Gödel ,
1Wu ,
min 1 ,
s
g
wu
a bJ a b
a b
a bJ a b
b a b
a bJ a b
a b a b
≤=
>
≤=
>
≤=
− >
• Satisfy the classical modus ponens for any T-norm T
Selection of Fuzzy Implications
• Modus Ponens: Some results for diff. J-T pairs
Selection of Fuzzy Implications
• Modus Tollens: Some results for diff. J-T pairs
Selection of Fuzzy Implications
• Hypothetical Syllogism: results for diff. J-T pairs