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