View
235
Download
0
Embed Size (px)
Citation preview
PART 3Operations on fuzzy sets
1. Fuzzy complements2. Fuzzy intersections3. Fuzzy unions4. Combinations of operations5. Aggregation operations
FUZZY SETS AND
FUZZY LOGICTheory and Applications
Fuzzy complements
• Axiomatic skeleton
Axiom c1.
Axiom c2.
(boundary conditions).
For all if , then (monotonicity).
01 and 10 cc
]1,0[, ba ba )()( bcac
Fuzzy complements
• Desirable requirements
Axiom c3.
Axiom c4.
c is a continuous function.
c is involutive, which means that for each
aacc ))((.]1,0[a
Fuzzy complements
• Theorem 3.1Let a function satisfy Axioms c2 and c4. Then, c also satisfies Axioms c1 and c3. Moreover, c must be a bijective function.
]1 ,0[]1 ,0[: c
Fuzzy complements
Fuzzy complements
Fuzzy complements
• Sugeno class
• Yager class
). ,1( where,1
1)(
a
aac
). ,0( where,)1()( /1 waac www
Fuzzy complements
Fuzzy complements
• Theorem 3.2Every fuzzy complement has at most one equilibrium.
Fuzzy complements
• Theorem 3.3
. iff
iff
c
c
eaaca
eaaca
Assume that a given fuzzy complement c has an equilibrium ec , which by Theorem 3.2 is unique. Then and
Fuzzy complements
• Theorem 3.4
• Theorem 3.5
If c is a continuous fuzzy complement, then c has a unique equilibrium.
If a complement c has an equilibrium ec , then
. ccd ee
Fuzzy complements
Fuzzy complements
• Theorem 3.6For each , , that is, when the complement is involutive.
]1 ,0[a aaccacad ))(( iff )(
Fuzzy complements
• Theorem 3.7
(First Characterization Theorem of Fuzzy Complements). Let c be a function from [0, 1] to [0, 1]. Then, c is a fuzzy complement (involutive) iff there exists a continuous function from [0, 1] to R such that , is strictly increasing, and
for all
0)0( g
))()1(()( 1 agggac
].1 ,0[a
g g
Fuzzy complements
• Increasing generators
Sugeno:
Yager:
.1for 1ln1
aag
.0for waag ww
Fuzzy complements
• Theorem 3.8 (Second Characterization Theorem of Fuzzy complements).
Let c be a function from [0, 1] to [0, 1]. Then c is a fuzzy complement iff there exists a continuous function from [0, 1] to R such that , is strictly decreasing, and
for all .
01 ff
afffac 01
]1 ,0[a
f
Fuzzy complements
• Decreasing generators
Sugeno:
Yager: .0 where,1)( waaf w
.1 where
),1ln(1
)1ln()(
aaaf
Fuzzy intersections: t-norms
• Axiomatic skeleton
Axiom i1.
Axiom i2.
(boundary condition).
implies (monotonicity).
aai 1,
db daibai ,,
Fuzzy intersections: t-norms
• Axiomatic skeleton
Axiom i3.
Axiom i4.
(commutativity).
(associativity).
abibai ,,
dbaiidbiai ,,,,
Fuzzy intersections: t-norms
• Desirable requirements
Axiom i5
Axiom i6
Axiom i7
is a continuous function (continuity).
(subidempotency).
implies
(strict monotonicity).
i
aaai ,
2121 and bbaa
),(),( 2211 baibai
Fuzzy intersections: t-norms
• Archimedean t-norm:
A t-norm satisfies Axiom i5 and i6.
• Strict Archimedean t-norm:
Archimedean t-norm and satisfies Axiom i7.
Fuzzy intersections: t-norms
• Frequently used t-norms
otherwise. 0
1 when
1 when
) ,( :onintersecti Drastic
)1 ,0(maxi :difference Bounded
.) ,( :product Algebraic
). ,min() ,( :onintersecti Standard
ab
ba
bai
baa, b
abbai
babai
Fuzzy intersections: t-norms
Fuzzy intersections: t-norms
Fuzzy intersections: t-norms
• Theorem 3.9
• Theorem 3.10
The standard fuzzy intersection is the only idempotent t-norm.
For all ,
where denotes the drastic intersection.
]1 ,0[, ba
, ,min , ,min babaibai
mini
Fuzzy intersections: t-norms
• Pseudo-inverse of decreasing generator
The pseudo-inverse of a decreasing generator , denoted by , is a function from R to [0, 1] given by
where is the ordinary inverse of .
f )1(f
f) ),0((for
)]0( ,0[for
)0 ,(for
0
)(
1
)( 1)1(
fa
fa
a
afaf
)1(f
Fuzzy intersections: t-norms
• Pseudo-inverse of increasing generator
The pseudo-inverse of a increasing generator , denoted by , is a function from R to [0, 1] given by
where is the ordinary inverse of .
g )1(g
g) ),1((for
)]1( ,0[for
)0 ,(for
1
)(
0
)( 1)1(
ga
ga
a
agag
)1(g
Fuzzy intersections: t-norms
• Lemma 3.1
Let be a decreasing generator. Then a function defined by
for any is an increasing generator
with , and its pseudo-inverse
is given by
for any R.
fg
)()0()( affag ]1 ,0[a)0()1( fg )1(g
a))0(()( )1()1( affag
Fuzzy intersections: t-norms
• Lemma 3.2
Let be a increasing generator. Then a function defined by
for any is an decreasing generator
with , and its pseudo-inverse
is given by
for any R.
gf
)()1()( aggaf ]1 ,0[a)1()0( gf )1(f
a))1(()( )1()1( aggaf
Fuzzy intersections: t-norms
• Theorem 3.11 (Characterization Theorem of t-Norms).
Let be a binary operation on the unit interval. Then, is an Archimedean t-norm iff there exists a decreasing generator
such that
for all .
f
]1 ,0[ , ba
))()(() ,( )1( bfaffbai
ii
Fuzzy intersections: t-norms
• [Schweizer and Sklar, 1963]
.))1 ,0(max(
otherwise.
]1 ,0[2when
0
)1(
)2(
))()(() ,(
) ,1( where
]1 ,0[ where
)0 ,( where
0
)1(
1
)(
).0( 1)(
1
1
)1(
)1(
p1)1(
pPp
ppppp
ppp
pppp
p
pp
ba
baba
baf
bfaffbai
z
z
z
zzf
paaf
Fuzzy intersections: t-norms
• [Yager, 1980f]
).])1()1([ ,1min(1
otherwise.]1 ,0[)1()1(
when
0
))1()1((1
))1()1((
))()(() ,(
) ,1( where
]1 ,0[ where
0
1)(
),0( )1()(
1
1
)1(
)1(
1)1(
www
ww
www
www
wwww
w
w
ww
ba
baba
baf
bfaffbai
z
zzzf
waaf
Fuzzy intersections: t-norms
• [Frank, 1979]
.1
)1)(1(1log
)1(
)1)(1()1(1log
)1(
)1)(1(ln
))()(( ) ,(
).)1(1(log)(
),1 ,0( 1
1ln)(
2
2)1(
)1(
)1(
s
ss
s
sss
s
ssf
bfaffbai
eszf
sss
saf
ba
s
ba
s
ba
s
ssss
zss
a
s
Fuzzy intersections: t-norms
• Theorem 3.12
Let denote the class of Yager t-norms.
Then,
for all , where the lower and upper bounds are obtained for and
,respectively.
wi
) ,min() ,() ,(min babaibai w
]1 ,0[ , ba0w w
Fuzzy intersections: t-norms
• Theorem 3.13
Let be a t-norm and be a
function such that is strictly increasing
and continuous in (0, 1) and
Then, the function defined by
for all ,where denotes the pseudo-inverse of , is also a t-norm.
i ]1 ,0[]1 ,0[: gg
.1)1( ,0)0( gggi
)))( ),((() ,( )1( bgagigbai g
]1 ,0[ , bag
)1(g
Fuzzy unions: t-conorms
• Axiomatic skeleton
Axiom u1.
Axiom u2.
).( )0 ,( onditionboundary caau
).( ) ,() ,( implies tymonotonicidaubaudb
Fuzzy unions: t-conorms
• Axiomatic skeleton
Axiom u3.
Axiom u4.
).( ) ,() ,( itycommutativabubau
).( ) ), ,(()) ,( ,( ityassociativdbauudbuau
Fuzzy unions: t-conorms
• Desirable requirements
Axiom u5.
Axiom u6.
Axiom u7.
).(function continuous a is continuityu
).(s ) ,( tencyuperidempoaaau
).( )()(
implies and
2211
2121
otonicitystrict mon, bau, bau
bbaa
Fuzzy unions: t-conorms
• Frequently used t-conorms
otherwise.
0when
0when
1
) ,( :union Drastic
). ,1min() ,( :sum Bounded
.) ,( :sum Algebraic
). ,max() ,( :union Standard
]1 ,0[ , allfor
a
b
b
a
bau
babau
abbabau
babau
ba
Fuzzy unions: t-conorms
Fuzzy unions: t-conorms
Fuzzy unions: t-conorms
• Theorem 3.14
The standard fuzzy union is the only idempotent t-conorm.
Fuzzy unions: t-conorms
• Theorem 3.15
For all
],1 ,0[ , ba
). ,() ,() ,max( max baubauba
Fuzzy unions: t-conorms
• Theorem 3.16 (Characterization Theorem of t-Conorms).
Let u be a binary operation on the unit interval. Then, u is an Archimedean t-conorm iff there exists an increasing generator such that
for all ].1 ,0[ , ba
))()(() ,( )1( bgaggbau
Fuzzy unions: t-conorms
• [Schweizer and Sklar, 1963]
.)1)1()1( ,0max(1
otherwise.
]1 ,0[)1()1(2when
1
]1)1()1[(1
))1(1)1(1() ,(
) ,1(when
]1 ,0[when
1
)1(1)(
).0( ) 1(1)(
1
1
)1(
1)1(
ppp
ppppp
ppp
p
p
p
pp
ba
baba
bagbau
z
zzzg
paag
Fuzzy unions: t-conorms
• [Yager, 1980f]
).)( ,1min(
)() ,(
) ,1(when
]1 ,0[when
1)(
),0( )(
1
)1(
1)1(
www
wwww
w
w
ww
ba
bagbau
z
zzzg
waag
Fuzzy unions: t-conorms
• [Frank, 1979]
.1
)1)(1(1log1) ,(
),)1(1(log1)(
)1 ,0( 1
1ln)(
11
)1(
1
s
ssbau
eszg
sss
sag
ba
ss
zss
a
s
Fuzzy unions: t-conorms
• Theorem 3.17
Let uw denote the class of Yager t-conorms.
for all where the lower and upper bounds are obtained for ,
respectively.
) ,() ,() ,max( max baubauba w
]1 ,0[ , ba
0 and ww
Fuzzy unions: t-conorms
• Theorem 3.18
Let u be a t-conorm and let be
a function such that is strictly increaning
and continuous in (0, 1) and .
Then, the function defined by
for all is also a t-conorm.
]1 ,0[]1 ,0[: g
g
1)1( ,0)0( gggu
]1 ,0[ , ba
)))( ),((() ,( )1( bgagugbau g
Combinations of operators
• Theorem 3.19
The triples
〈 min, max, c 〉 and 〈 imin, umax, c 〉 are dual
with respect to any fuzzy complement c.
Combinations of operators
• Theorem 3.20
Given a t-norm i and an involutive fuzzy complement c, the binary operation u on [0, 1] defined by
for all is a t-conorm such that
〈 i, u, c 〉 is a dual triple.
]1 ,0[ , ba
)))( ),((() ,( bcacicbau
Combinations of operators
• Theorem 3.21
Given a t-conorm u and an involutive fuzzy complement c, the binary operation i on
[0, 1] defined by
for all is a t-norm such that
〈 i, u, c 〉 is a dual triple.
]1 ,0[ , ba
)))( ),((() ,( bcacucbai
Combinations of operators
• Theorem 3.22
Given an involutive fuzzy complement c and an increasing generator of c, the
t-norm and t-conorm generated by are dual with respect to c.
g
g
Combinations of operators
• Theorem 3.23
Let 〈 i, u, c 〉 be a dual triple generated by Theorem 3.22. Then, the fuzzy operations i, u, c satisfy the law of excluded middle and the law of contradiction.
Combinations of operators
• Theorem 3.24
Let 〈 i, u, c 〉 be a dual triple that satisfies the law of excluded middle and the law of contradiction. Then, 〈 i, u, c 〉 does not satisfy the distributive laws.
Aggregation operations
• Axiomatic requirements
Axiom h1.
Axiom h2.
). ( 1)1 ..., ,1 ,1( and 0)0 ..., ,0 ,0( onditionsboundary chh
arguments. its allin is is,that
; )()(
then
, allfor if , allfor ]10[such that
tuples- of ,... , , and ,... , ,pair any For
2121
2121
ingincreasmonotonic h
, ..., b, bbh, ..., a, aah
NibaNi, , ba
nbbbaaa
nn
niinii
nn
Aggregation operations
• Axiomatic requirements
Axiom h3.
function. is continuoush
Aggregation operations
• Additional requirements
Axiom h4.
Axiom h5.
.on n permutatioany for
)()(
is, that arguments; its allin function a is
)()2()1(21
n
npppn
Np
, ..., a, aah, ..., a, aah
symmetrich
].1 ,0[ allfor
)(
is, that function; an is
a
a aa, a, ...,h
idempotenth
Aggregation operations
• Theorem 3.25
.
121
21212211
n
N allfor 0 where
,)(
Then, .N allfor 1] [0, where
)()()(
property theand h2,
Axiomh1, Axiom satisfieshat function t a be R1] [0,:Let
ni
n
iiin
niiii
nnnn
iw
aw, ..., a, aah
ib, a, ba
, ..., b, bbh, ..., a, aahb, ..., ab, abah
h
Aggregation operations
• Theorem 3.26
.
1121
212111
N allfor 1] [0, where
),) ,min( ..., ), ,max(min()(
Then, .N allfor 0) ..., 0, , ...,0, (0,)( where
)())( (
))( ),(max()) ,max() ,(max(
property theand h3,
Axiomh1, Axiom satisfieshat function t a be 1] [0,1] [0,:Let
ni
nnn
niii
iiiii
nnnn
n
iw
awaw, ..., a, aah
iahah
ahahh
, ..., b, bbh, ..., a, aahba, ..., bah
h
Aggregation operations
• Theorem 3.27
).min()(
such that 1] [0,..., numbers
exist thereThen, .N allfor 1) ..., 1, , ...,1, (1,)( where
0)0( and )()( (ab)
))( ),(min()) ,min() ,(min(
property theand h3,
Axiomh1, Axiom satisfieshat function t a be 1] [0,1] [0,:Let
21
2121
21
212111
nαn
ααn
n
niii
iiii
nnnn
n
, ..., a, aa, ..., a, aah
α,α,α
iahah
hbhahh
, ..., b, bbh, ..., a, aahba, ..., bah
h
Aggregation operations
• Theorem 3.28
1]. [0,any for
otherwise
]1 ,[ where
] [0, where
)min(
) ,max(
) ,(
such that 1] [0, exists thereThen,
.idempotent and continuous be operation norm aLet
a, b
a, b
a, b
a, b
ba
bah
h
Exercise 3
• 3.6
• 3.7
• 3.13
• 3.14