fuzzy rendszerek i
DESCRIPTION
1 / 1. Fuzzy Rendszerek I. Dr. K óczy László. An Example. 1 / 2. A class of students (E.G. M.Sc. Students taking „Fuzzy Theory”) The universe of discourse : X. “Who does have a driver’s licence?” A subset of X = A (Crisp) Set (X) = CHARACTERISTIC FUNCTION. - PowerPoint PPT PresentationTRANSCRIPT
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Fuzzy Rendszerek I.
Dr. Kóczy László
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An Example• A class of students
(E.G. M.Sc. Students taking „Fuzzy Theory”)
• The universe of discourse: X
• “Who does have a driver’s licence?”• A subset of X = A (Crisp) Set (X) = CHARACTERISTIC FUNCTION
• “Who can drive very well?” (X) = MEMBERSHIP FUNCTION
1 0 1 1 0 1 1
0.7 0 1.0 0.8 0 0.4 0.2
FUZZY SET
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History of fuzzy theory
• Fuzzy sets & logic: Zadeh 1964/1965-• Fuzzy algorithm: Zadeh 1968-(1973)-• Fuzzy control by linguistic rules: Mamdani & Al. ~1975-• Industrial applications: Japan 1987- (Fuzzy boom), Korea
Home electronicsVehicle controlProcess controlPattern recognition & image processingExpert systemsMilitary systems (USA ~1990-)Space research
• Applications to very complex control problems: Japan 1991-E.G. helicopter autopilot
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An application exampleOne of the most interesting applications of fuzzy computing: “FOREX” system.
1989-1992, Laboratory for International Fuzzy Engineering Research (Yokohama, Japan) (Engineering – Financial Engineering)
To predict the change of exchange rates (FOReign EXchange)
~5600 rules like:“IF the USA achieved military successes on the past day [E.G. in the Gulf War] THEN ¥/$ will slightly rise.”
Inputs
FOREX Predictio
nFuzzy Inference Engine
(Observations)
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Another Example
What is fuzzy here?
- What is the tendency of the ¥/$ exchange rate?“It’s MORE OR LESS falling” (The general tendency is “falling”, there’s no big interval of rising, etc.)
- What is the current rate?Approximately 88 ¥/$ Fuzzy number
- When did it first cross the magic 100 ¥/$ rate? SOMEWHEN in mid 1995
1993 1994 1995 Time
¥/$
100
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A complex problem
Many components, very complex system. Can AI system solve it? Not, as far as we know. But WE can.
Our car, save fuel, save time, etc.
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Definitions• Crisp set:
• Convex set:A is not convex as aA, cA, butd=a+(1-)c A, [0, 1].B is convex as for every x, yB and[0, 1] z=x+(1-)y B.
• Subset:
• x• y
c •
• a
• b• d
Crisp set A
B
aAbA
• x• y
A
BIf xA then also xB.
AB
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Definitions• Equal sets:
If AB and BA then A=B if not so AB.
• Proper subset:If there is at least one yB such that yA then AB.
• Empty set: No such x.
• Characteristic function:A(x): X{0, 1}, where X the universe.0 value: x is not a member,1 value: x is a member.
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DefinitionsA={1, 2, 3, 4, 5, 6}
• Cardinality: |A|=6.• Power set of A:
P (A)={{}=Ø, {1}, {2}, {3}, {4}, {5}, {6}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 5}, {2, 4, 6}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {2, 3, 4, 5}, {2, 3, 4, 6}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6}, {2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6}}.|P (A)|=2|6|=64.
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Definitions• Relative complement or difference:
A–B={x | xA and xB}B={1, 3, 4, 5}, A–B={2, 6}.C={1, 3, 4, 5, 7, 8}, A–C={2, 6}!
• Complement: where X is the universe.Complementation is involutive:Basic properties:
• Union:AB={x | xA or xB}
For
AXA AA
X
,X
n
1iiii i somefor Ax|xA Ii|A
XAA
AA
XXA
(Law of excluded middle)
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Definitions• Intersection:
AB={x | xA and xB}.
For
• More properties: Commutativity: AB=BA, AB=BA. Associativity: ABC=(AB)C=A(BC),
ABC=(AB)C=A(BC). Idempotence: AA=A, AA=A. Distributivity:
A(BC)=(AB)(AC),A(BC)=(AB)(AC).
n
1iiii i allfor Ax|xA Ii|A
AA
AXA
A(Law of contradiction)
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Definitions• More properties (continued):
DeMorgan’s laws:
• Disjoint sets: AB=.
• Partition of X:
BABA
BABA
Iiiiii XA ,AA ,Ii|Ax
21
xNi|A
AA
AX
6i
ji
6
1ii
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Summarize propertiesInvolution
Commutativity AB=BA, AB=BA
AssociativityABC=(AB)C=A(BC),ABC=(AB)C=A(BC)
DistributivityA(BC)=(AB)(AC),A(BC)=(AB)(AC)
Idempotence AA=A, AA=A
Absorption A(AB)=A, A(AB)=A
Absorption of complement
Abs. by X and AX=X, A=
Identity A=A, AX=A
Law of contradiction
Law of excl. middle
DeMorgan’s laws
AA
BABAA
BABAA
AA
XAA
BABABABA
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Membership functionCrisp set Fuzzy set
Characteristic function Membership function
A:X{0, 1} A:X[0, 1]
10x 2A51
1
17x1017x71
10x55x2.0
17x5x0
B 2AC
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Membership function (2)• More general Idea:
L-Fuzzy set, A: XL, L has a linear (full) or partial ordering.
• Vector valued fuzzy set:A: x[0, 1]n=[0,1]…[0,1] (n times).
• Fuzzy measure:g: P (x)[0, 1]Degree of belonging to a crisp subset of the universe.
Example:’teenager’=13..19 years old, ‘twen’=20..29 years old,‘in his thirties’=30..39 years old, etc.X={Years of men’s possible age}‘teenager’Xg(‘Susan is a twen’)=0.8,g(‘Susan is a teenager’)=0.2,g(‘Susan is in her thirties’)=0.1.Best guess has highest measure.
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Definitions• Fuzzy set of type 2: 1,0 P
~: X
• Level N:
0
X1 X2 X
1 1
2A
P~P~
1,0)x(~: P 1-k 1k)x(~~)x(~ )P(P P 1-kk
x)x(~ P0
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Definitions• Interval valued fuzzy set:
even
• C(Y) is the family of convex subsets of Y, i.e.
P(Y)• C([0,1]) is the family of all intervals
• e.g.:
1,0~X: P 1,0CX:
)x(
0
X
1
]0.1,9.0[)x(
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Definitions• Relation of various classes of sets discussed
Crisp Set
Fuzzy Set
Interval V’D
Type N-1
Type 2
Type N
Vector V’D
L-Fuzzy
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Some basic concepts of fuzzy setsEle-
mentsInfant Adult Young Old
5 0 0 1 0
10 0 0 1 0
20 0 .8 .8 .1
30 0 1 .5 .2
40 0 1 .2 .4
50 0 1 .1 .6
60 0 1 0 .8
70 0 1 0 1
80 0 1 0 1
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Some basic concepts of fuzzy sets
• Support: supp(A)={x | A(x)>0}. supp:Infant=0, so supp(Infant)=0.If |supp(A)|<, A can be defined
A=1/x1+ 2/x2+…+ n/xn.
• Kernel (Nucleus, Core): Kernel(A)={x | A(x)=1}.
xx~ PP
n
1iii x/A
xA x/xA
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Definitions• Height:
• height(old)=1 height(infant)=0– If height(A)=1 A is normal– If height(A)<1 A is subnormal– height(0)=0
• -cut: Strong Cut:
•
– Kernel:– Support:
• If A is subnormal, Kernel(A)=0
–
))x((sup))x((max)A(Height Ax
Ax
})x(|x{A A
})x(|x{A A }20,10,5{Young 8.0 }10,5{Young 8.0
}1)x(|x{A A1 }0)x(|x{A A0
IFAA
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Definitions
• Level set of A:
• Convex fuzzy set:
– A is convex if for every x,yX and [0,1]:
– All sets on the previous figure are convex FS’s
}Xxsomefor)x(|{ AA
nX
))y(),x(min()y)1(x( AAA
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Definitions• Nonconvex fuzzy set:
• Convex fuzzy set over R 2
=0.2=0.4
=0.6=0.8
Kernel
x1
x2
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Definitions• Fuzzy cardinality of FS A:
Fuzzy number
for all A
A~
AA~
2/13/3.04/6.05/4.06/2.07/1.0dl~
O
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Definitions• Fuzzy number: Convex and normal fuzzy set of
– Example 1:
height(N(r))=1
for any r1, r2: N(r*)= N(r1+(1- )r2) min(N(r1), N(r2))
– Example 2: “Approximately equal to 6”
0r2
1
N
r*r1
otherwise0
]9,3[rif3
6r1)r(N
09
1
N(r)
63
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Definitions• Flat fuzzy number:
There is a,b (ab, a,b) N(r)=1 IFF r[a,b]
(Extension of ‘interval’)• Containment (inclusion) of fuzzy set
AB IFF A(x) B(x)– Example: Old Adult
• Equal fuzzy sets
A=B IFF AB and AB If it is not the case: AB
• Proper subset
AB IFF AB and A B– Example: Old Adult
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Definitions• Extension principle:
How to generalize crisp concepts to fuzzy?Suppose that X and Y are crisp sets
X={xi}, Y={yi} and f: XY
If given a fuzzy set
The latter is understood that if for
xA P~
ISYPBTHENxAn
iii )(
~/
1
n
iii
n
iii xfxfAfB
11
//
THENjxFORxfy nij
iji |
ijji yxfxy |max
0 iij yTHENxIF
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Definitions– Example
X={a,b,c}
Y={d,e,f}
• Arithmetics with fuzzy numbers:Using extension principle
a+b=c (a,b,c )
A+B=C
dc
fb
da
f
fdY
cbaX
Af
A
/5./8.
/8./5./1.
?
~~,
CISWHAT
CEVENPBA
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Example
nbaban BAba
C
|,minmax,
0
1 C
0 1 2 3 4 5 6 7 8 9 10 11 12
A B
n a b min(A,B)3 0 3 0
4 0
1
4
3
0
0
5 0
1
2
5
4
3
0
0.5
0
6 0
1
2
3
6
5
4
3
0
0.5
0.5
7 0
1
...
7
6
0
0.5
n a b min(A,B)7 2
3
4
5
4
3
1
0.5
0
8 1
2
3
4
7
6
5
4
0
0.5
0.5
0
9 2
3
4
7
6
5
0
0.5
0
10 3
4
7
6
0
0
11 4 7 0
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Definitions
• Fuzzy set operations defined by L.A. Zadeh in 1964/1965
• Complement: • Intersection:• Union:
x1x AA x,xminx BABA x,xmaxx BABA
(x):
B,Ax1B,Ax0
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Definitions
A B A AB AB 1-A min max
0 0 1 0 0 1 0 00 1 1 0 1 1 0 11 0 0 0 1 0 0 11 1 0 1 1 0 1 1
This is really a generalization of crisp set op’s!
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Classical (Two valued) logic• Classical (two valued logic):
Any logic – Means of a reasoning propositions: true (1) or false (0). Propositional logic uses logic variables, every variable stands for a proposition. When combining LV’s new variables (new propositions) can be constricted.Logical function:f: v1, v2, …, vn vn+1
• different logic functions of n variables exist. E.G. if n=2, 16 different logic functions (see next page).
n22
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Logic functions of two variables v2
v1
1100
1010
Adopted name Symbol Other names used
w1 0000 Zero fn. 0 Falsum
w2 0001 Nor fn. v1v2 Pierce fn.
w3 0010 Inhibition v1>v2 Proper inequality
w4 0011 Negation v2 Complement
w5 0100 Inhibition v1<v2 Proper inequality
w6 0101 Negation v1 Complement
w7 0110 Excl. or fn. v1v2 Antivalence
w8 0111 Nand fn. v1v2 Sheffer Stroke
w9 1000 Conjunction v1v2 And function
w10 1001 Biconditional v1v2 Equivalence
w11 1010 Assertion v1 Identity
w12 1011 Implication v1v2 Conditional, ineq.
w13 1100 Assertion v2 Identity
w14 1101 Implication v1v2 Conditional, ineq.
w15 1110 Disjunction v1v2 Or function
w16 1111 One fn. 1 Verum
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Definitions
• Important concern:How to express all logic fn’s by a few logic primitives (fn’s of one or two lv’s)?
A system of lp’s is (functionally) complete if all LF’s can be expressed by the FNs in the system.
A system is a base system if it is functionally complete and when omitting any of its elements the new system isn’t functionally complete.
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Definitions• A system of LFN’s is functionally complete if:
– At least one doesn’t preserve 0– At least one doesn’t preserve 1– At least one isn’t monotonic– At least one isn’t self dual– At least one isn’t linear
• Example: AND , NOT
ONLYBY
EXPRESSEDBECANNOTBA
BABABABA
A
A
A
01,10,01,10
11
00
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Definitions• Importance of base systems, and functionally
complete systems.Digital engineering:Which set of primitive digital circuits is suitable to construct an arbitrary circuit?
• Very usual: AND, OR, NOT (Not easy from the point of view of technology!)
• NAND (Sheffer stroke)
• NOR (Pierce function)
• NOT, IMPLICATION (Very popular among logicians.)
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Definitions• Logic formulas:
– E.G.: Let’s adopt +,-,* as a complete system. Then:
– Similarly with -,, etc.• There are infinitely many ways to describe a LF in an
equivalent way– E.G.: A, , A+A, A*A, A+A+A, etc.
• Canonical formulas, normal form– DCF– CCF– Logic formulas with identical truth values are named
equivalent
A
BABABABA
BABAABBA
• If A and B are LFs, then A+B, A*B are LFs• There are no other LFsv
• 0 and 1 are LFs• If v is a LF is a LF
a=b
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Definitions• Always true logic formulas:
Tautology: AA+A
• Always false logic formulas:Contradiction: A
• Various forms of tautologies are used for didactive inference = inference rules– Modus Ponens:
(a*(ab))b
– Modus Tollens:
– Hypothetical Syllogism:((a b)*(b c)) (a c)
– Rule of Substitution:If in a tautology any variable is replaced by a LF
then it
remains a tautology.
A
aba*b
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Definitions
• These inference rules form the base of expert systems and related systems (even fuzzy control!)
• Abstract algebraic model:Boolean Algebra
B=(B,+,*,-)B has at least two different elements (bounds): 0 and 1+ some properties of binary OPs “+” and “*”, and unary OP “-”.
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Properties of boolean algebras
B1. Idempotence a+a=a, a·a=a
B2. Commutativity a+b=b+a
B3. Associativity (a+b)+c=a+(b+c),
(a·b)·c=a·(b·c)
B4. Absorption a+(a·b)=a, a·(a+b)=a
B5. Distributivity a·(b+c)=(a·b)+(a·c),
a+(b·c)=(a+b)·(a+c)
B6. Universal bounds a+0=a, a+1=1
a·1=a, a·0=0
B7. Complementarity a+a=1, a·a=0, 1=0
B8. Involution (a+b)= a · b
B9.Dualization (a·b)= a +b
Lattice
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Definitions
• Correspondences defining isomorphisms between set theory, boolean algebra and propositional logic
Set theory
Boolean algebra
Propositional logic
P(X) B F(V)
+
*
- - -
X 1 1
0 0
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Definitions• Isomorphic structure of crisp set and logic
operations = boolean algebra
• Structure of propositions:x is P
–
– x2=Dr.Kim P(x2)=FALSE!
B.A.
Lattice
TRUEPREDICATE
PSUBJECT
x
190cm above isKóczy Dr.
1
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Definitions
• Quantifiers:– (x) P(x): There exists an x such
that x is P– (x) P(x): For all x, x is P– (!x) P(x): x and only one x such that x is P
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Definitions• Two valued logic questioned since B.C.
Three valued logic includes indeterminate value: ½Negation: 1-a, differ in these logics.
• Examples:
ab Łukasiewicz
Bochvar
Kleene
Heyting
Reichenbach
00 0011 0011 0011 0011 0011
0½ 0½1½ ½½½½ 0½1½ 0½10 0½1½
01 0110 0110 0110 0110 0110
½0 0½½½ ½½½½ 0½½½ 0½00 0½½½
½½ ½½11 ½½½½ ½½½½ ½½11 ½½11
½1 ½11½ ½½½½ ½11½ ½11½ ½11½
10 0100 0100 0100 0100 0100
1½ ½1½½ ½½½½ ½1½½ ½1½½ ½1½½
11 1111 1111 1111 1111 1111
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Definitions
• No difference from classical logic for 0 and 1.
But: are not true!
• Quasi tautology: doesn’t assume 0. Quasi contradiction: doesn’t assume 1.
• Next step?
1aa ,0aa
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N-valued logic• N-valued logic:
Degrees of truth(Lukasiewicz, ~1933)
1,
1
2,...,
1
1,0Tn n
n
n
b-a-1baa)-bmin(1,1ba
ba,maxbaba,minba
a-1a
LOGIC
PRIMITIVES
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Definitions• Ln n=2, … ,
(0) Classical LogicIf T is extended to [0,1] we obtain (T1
) L1 with continuum
truth degrees
• L1 is isomorphic with fuzzy set
It is enough to study one of them, it will reveal all the facts above the other.
• Fuzzy logic must be the foundator of approximate reasoning, based on natural language!
Rational truth values
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Fuzzy Proportion• Fuzzy proportion: X is P
‘Tina is young’, where:‘Tina’: Crisp age, ‘young’: fuzzy predicate.
Fuzzy sets expressing linguistic terms for ages
Truth claims – Fuzzy sets over [0, 1]
• Fuzzy logic based approximate reasoning is most important for applications!