Jan. 2011Computer Architecture, The Arithmetic/Logic UnitSlide 1 Part III The Arithmetic/Logic Unit.
PART 4 Fuzzy Arithmetic
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Transcript of PART 4 Fuzzy Arithmetic
PART 4Fuzzy Arithmetic
1. Fuzzy numbers2. Linguistic variables3. Operations on intervals4. Operations on fuzzy numbers5. Lattice of fuzzy numbers6. Fuzzy equations
FUZZY SETS AND
FUZZY LOGICTheory and Applications
Fuzzy numbers
• Three properties1) A must be a normal fuzzy set;
2) αA must be a closed interval for every
3) the support of A, 0+A, must be bounded.
A is a fuzzy set on R.
];1 ,0(
Fuzzy numbers
Fuzzy numbers
• Theorem 4.1
Let Then, A is a fuzzy number if and only if there exists a closed interval
such that
).(RFA
] ,[ ba
), ,(for
) ,(for
] ,[for
)(
)(
1
)(
bx
ax
bax
xr
xlxA
Fuzzy numbers
• Theorem 4.1 (cont.)
where is a function from that is
monotonic increasing, continuous from the right,
and such that ; is a
function from that is monotonic decreasing, continuous from the left, and such
that
l 1] [0, to) ,( a
) ,(for 0)( 1 xxl r1] [0, to) ,( b
) ,(for 0)( 2 xxr
Fuzzy numbers
Fuzzy numbers
Fuzzy numbers
• Fuzzy cardinality
Given a fuzzy set A defined on a finite universal set X, its fuzzy cardinality, , is a fuzzy number defined on N by the formula
for all
|~
| A
|)(||~
| AA
).(A
Linguistic variables
• The concept of a fuzzy number plays a fundamental role in formulating quantitative fuzzy variables.
• The fuzzy numbers represent linguistic concepts, such as very small, small, medium, and so on, as interpreted in a particular context, the resulting constructs are usually called linguistic variables.
Linguistic variables
• base variable
Each linguistic variable the states of which are expressed by linguistic terms interpreted as specific fuzzy numbers is defined in terms of a base variable, the values of which are real numbers within a specific range.
A base variable is a variable in the classical sense, exemplified by any physical variable (e.g., temperature, etc.) as well as any other numerical variable, (e.g., age, probability, etc.).
Linguistic variables
• Each linguistic variable is fully characterized by a quintuple (v, T, X, g, m).– v : the name of the variable.– T : the set of linguistic terms of v that refer to
a base variable whose values range over a universal set X.
– g : a syntactic rule (a grammar) for generating linguistic terms.
– m : a semantic rule that assigns to each linguistic term t T.
Linguistic variables
Operations on intervals
• Let * denote any of the four arithmetic operations on closed intervals: addition + , subtraction —, multiplication • , and division /. Then,
)].e/ ,/ ,/ ,/max(
),e/ ,/ ,/ ,/[min(
]d1 ,1[] ,[] ,/[] ,[
)],e , , ,max( ),e , , ,[min(] ,[] ,[
], ,[] ,[] ,[
], ,[] ,[] ,[
}, ,|{] ,[] ,[
bdbeada
bdbeada
ebaedba
bbdaeadbbdaeadedba
dbeaedba
ebdaedba
egdbfagfedba
Operations on intervals
• Properties
Let ].1 ,1[ ],0 ,0[ ], ,[ ], ,[ ], ,[ 212121 10ccCbbBaaA
).( )( .4
).(
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utivitysubdistribCABACBA
identityAAA
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ityassociativCBACBA
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itycommutativABBA
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Operations on intervals
). ( //
,
,
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: then, and If .7
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.)( then ], ,[ if e,Furthermor ).(
)( then , and every for 0 If .5
tymonotoniciinclusionFEBA
FEBA
FEBA
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FBEA
A/AA-A
CaBaCBaaaAvitydistributi
CABACBACcBbcb
Operations on fuzzy numbers
• First method
Let A and B denote fuzzy numbers. * denote any of the four basic arithmetic operations.
for any
Since is a closed interval for each
and A, B are fuzzy numbers, is
also a fuzzy number.
].1 ,0(BABA )(
].1 ,0(
.)(1] [0,αα
BABA
)( BA
BA
Operations on fuzzy numbers
• Second method
)].( ),(min[sup)B)(/(A
)],( ),(min[sup)B)((A
)],( ),(min[sup)B)((A
)],( ),(min[sup)B)((A
)],( ),(min[sup)B)((A
allfor
/yBxAz
yBxAz
yBxAz
yBxAz
yBxAz
z
yxz
yxz
yxz
yxz
yxz
R
Operations on fuzzy numbers
Operations on fuzzy numbers
Operations on fuzzy numbers
• Theorem 4.2
Let * { + , - , •, / }, and let A, B denote continuous fuzzy numbers. Then, the fuzzy set
A * B defined by
is a continuous fuzzy number.
)]( ),(min[sup)B)((A yBxAzyxz
Lattice of fuzzy numbers
• MIN and MAX
)].( ),(min[sup))( ,(
)],( ),(min[sup))( ,(
) ,max(
) ,min(
yBxAzBA
yBxAzBA
yxz
yxz
MAX
MIN
Lattice of fuzzy numbers
Lattice of fuzzy numbers
Lattice of fuzzy numbers
• Theorem 4.3
Let MIN and MAX be binary operations on R.
Then, for any , the following properties hold:
RCBA , ,
Lattice of fuzzy numbers
Lattice of fuzzy numbers
• Lattice
It also can be expressed as the pair , where is a partial ordering defined as:
MAXMIN , ,R ,R
intervals. closed are where
,)ax( iff
,)in( iff
:cuts-relevant theof in terms ordering partial
thedefine alsocan we],10( all and any for
)( iff
ely,alternativ or, )( iff
BA,
BBA, BA
ABA, BA
, αRA, B
BA, BBA
AA, BBA
m
m
MAX
MIN
Lattice of fuzzy numbers
].10( allfor
iff
have we,any for then
, and iff ][][
is, that way,
usual in the intervals closed of ordering partial thedefine weIf
)].(max ),(max[)(max
)],(min ),(min[)(min
Then,
22112121
2211
2211
,
BA BA
A, B
baba, bb, aa
, ba, baBA,
, ba, baBA, αα
αα
R
Fuzzy equations
• A + X = B
The difficulty of solving this fuzzy equation is caused by the fact that X = B - A is not the solution.
Let A = [a1, a2] and B = [b1, b2] be two closed intervals, which may be viewed as special fuzzy numbers. B - A = [b1- a2 , b2 - a1], then
Fuzzy equations
Let X = [x1, x2].
]. ,[
. iffsolution a hasequation the
. that required sit' interval,an bemust
.
.
,
,
]. ,[] ,[ Then,
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2211
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222
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222
111
212211
ababX
abab
xxX
abx
abx
bxa
bxa
bbxaxa
Fuzzy equations
Let αA = [αa1, αa2], αB = [αb1, αb2], and
αX = [αx1, αx2] for any . ]1 ,0(
]10(
22221111
2211
.
bygiven isequation fuzzy theof solution the
. implies (ii)
and ],10(every for (i)
:iffsolution a has
, α
ααββββαα
XX
X
ababababβα
, αabab
BXA
Fuzzy equations
• A . X = B
A, B are fuzzy numbers on R+. It’s easy to show that X = B / A is not a solution of the equation.
]10(
22221111
2211
.
bygiven isequation fuzzy theof solution the
.//// implies (ii)
and ],10(every for // (i)
:iffsolution a has
, α
ααββββαα
XX
X
ababababβα
, αabab
BXA
Exercise 4
• 4.1
• 4.2
• 4.5
• 4.6
• 4.9