PART 6 Fuzzy Logic 1. Classical logic 2. Multivalued logics 3. Fuzzy propositions 4. Fuzzy quantifiers 5. Linguistic hedges FUZZY SETS AND FUZZY LOGIC Theory and Applications
Dec 13, 2015
PART 6Fuzzy Logic
1. Classical logic2. Multivalued logics3. Fuzzy propositions4. Fuzzy quantifiers5. Linguistic hedges
FUZZY SETS AND
FUZZY LOGICTheory and Applications
6. Inference from conditional fuzzy propositions7. Inference from conditional and qualified propositions8. Inference from quantified propositions
FUZZY SETS AND
FUZZY LOGICTheory and Applications
33
Classical logic
• Inference rules Various forms of tautologies can be used for
making deductive inferences. They are referred to as inference rules. Examples :
.syllogism) cal(hypotheti )())()((
tollens),(modus ))((
ponens), (modus ))((
cacbba
abab
bbaa
44
Classical logic
• Existential quantifier Existential quantification of a predicate P(x) is
expressed by the form
"There exists an individual x (in the universal set X of the variable x) such that x is P". We have the following equality:
),()( xPx
)()()( xPxPxXx
V
55
Classical logic
• Universal quantifier Universal quantification of a predicate P(x) is
expressed by the form.
“For every individual x (in the universal set) x is P". Clearly, the following equality holds:
),()( xPx
( ) ( ) ( )x X
x P x P x
66
Classical logic
• General quantifier Q
The quantifier Q applied to a predicate P(x),
x X, as a binary relation
where α, β specify the number of elements of X for which P(x) is true or false, respectively. Formally,
|},| ,|){( Xβαα, βα, β NQ
.|}false is )(|{|
|,} trueis )(|{|
xPXx
xPXx
77
Multivalued logics
88
Multivalued logics
• n-valued logics For any given n, the truth values in these
generalized logics are usually labelled by rational numbers in the unit interval [0, 1]. The set Tn of truth values of an n-valued logic is thus defined as
These values can be interpreted as degrees of truth.
.11
1,
1
2 ,
1
2 ,
1
1 ,
1
00
n
n
n
n
nnnTn
99
Multivalued logics
Lukasiewicz uses truth values in Tn and defines the primitives by the following equations:
.||1
),1 ,1min(
), ,max(
), ,min(
,1
baba
abba
baba
baba
aa
1010
Multivalued logics
Lukasiewicz, in fact, used only negation and implication as primitives and defined the other logic operations in terms of these two primitives as follows:
).()(
,
,)(
abbaba
baba
bbaba
1111
Fuzzy propositions
• Unconditional and unqualified proposition The canonical form of fuzzy propositions of this
type, p, is expressed by the sentence
where V is a variable that takes values v from some universal set V, and F is a fuzzy set on V that represents a fuzzy predicate, such as tall, expensive, low, normal, and so on.
, is : Fp V
1212
Fuzzy propositions
Given a particular value of V (say, v), this value belongs to F with membership grade F(v). This membership grade is then interpreted as the degree of truth, T(p), of proposition p. That is,
for each given particular value v of variable V in proposition p. This means that T is in effect a fuzzy set on [0,1], which assigns the membership grade F(v) to each value v of variable V.
)()( vFpT
1313
Fuzzy propositions
1414
Fuzzy propositions
In some fuzzy propositions, values of variable V are assigned to individuals in a given set / . That is, variable V becomes a function V : / → V, where V ( i ) is the value of V for individual i in V. The canonical form must then be modified to the form
. where, is )( : IiFip V
Fp is :V
1515
Fuzzy propositions
• Unconditional and qualified proposition Propositions p of this type are characterized by
either the canonical form
or the canonical form
(8.8) , is } is Pro{ : PFp V
, is is : SFp V
1616
Fuzzy propositions
In general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v V by the equation
An example of a truth-qualified proposition is the proposition "Tina is young is very true."
.))(()( vFSpT
1717
Fuzzy propositions
1818
Fuzzy propositions
Let us discuss now probability-qualified propositions of the form (8.8). For any given probability distribution f on V, we have
and, then, the degree T(p) to which proposition p of the form (8.8) is true is given by the formula
;)()(} is Pro{
Vv
vFvfFV
))()(()(
Vv
vFvfPpT
1919
Fuzzy propositions
As an example, let variable V be the average daily temperature t in °F at some place on the Earth during a certain month. Then, the probability-qualified proposition
p : Pro { temperature t (at given place and time) is around 75 °F } is likely
may provide us with a meaningful characterization of one aspect of climate at the given place and time.
2020
Fuzzy propositions
2121
Fuzzy propositions
• Conditional and unqualified proposition Propositions p of this type are expressed by the
canonical form
where X, Y are variables whose values are in sets X, Y, respectively, and A, B are fuzzy sets on X, Y, respectively.
, is then , is If : BAp YX
2222
Fuzzy propositions
These propositions may also be viewed as propositions of the form
where R is a fuzzy set on X x Y that is determined for each x X and each y Y by the formula
where J denotes a binary operation on [0, 1] representing a suitable fuzzy implication.
, is , RYX
,)]( ),([) ,( yBxAyxR J
2323
Fuzzy propositions
Here, let us only illustrate the connection for one particular fuzzy implication, the Lukasiewicz implication
This means, for example, that T(p) = 1 when X = x1 and Y = y1; T(p) = .7 when X = x2 and Y = y1 and so on.
).1 ,1min() ,( baba J
.,1,5.
,1,7.,1,1Then
.15. and 18.1.Let
2313
22122111
21321
yxyx
yxyxyxyxR
yyBxxxA
2424
Fuzzy propositions
• Conditional and unqualified proposition Propositions of this type can be characterized by
either the canonical form
or the canonical form
where Pro {X is A | Y is B} is a conditional probability.
, is is then , is If : SBAp YX
, is } is | is Pro{ : PBAp YX
2525
Fuzzy quantifiers
• First Kind - Ⅰ There are two basic forms of propositions that
contain fuzzy quantifiers of the first kind. One of them is the form
where V is a variable that for each individual i in a given set / assumes a value V(i), F is a fuzzy set defined on the set of values of variable V, and Q is a fuzzy number on R.
, is such that in are There: F(i)Ii'sQp V
2626
Fuzzy quantifiers
Any proposition p of this form can be converted into another proposition, p', of a simplified form,
where E is a fuzzy set on a given set / that is defined by the composition
, are There:' E'sQp
. allfor ))(()( IiiFiE V
2727
Fuzzy quantifiers
For example, the proposition
p : "There are about 10 students in a given class whose fluency in English is high“
can be replaced with the proposition
p’ : "There are about 10 high-fluency English-speaking students in a given class."
Here, E is the fuzzy set of "high-fluency English-
speaking students in a given class."
2828
Fuzzy quantifiers
Proposition p' may be rewritten in the form
where W is a variable taking values in R that represents the scalar cardinality, W = |E|,
and,
, is :' Qp W
IiIi
iFiEE ))(()(|| V
|).(|)'()( EQpTpT
2929
Fuzzy quantifiers
Example :
p : There are about three students in / whose fluency in English, V( i ), is high.
Assume that / = {Adam, Bob, Cathy, David, Eve}, and V is a variable with values in the interval [0, 100] that express degrees of fluency in English.
3030
Fuzzy quantifiers
3131
Fuzzy quantifiers
• First Kind - Ⅱ Fuzzy quantifiers of the first kind may also
appear in fuzzy propositions of the form
where V1, V2 are variables that take values from sets V1, V2, respectively, / is an index set by which distinct measurements of variables V1,V2 are identified (e.g., measurements on a set of individuals or measurements at distinct time instants), Q is a fuzzy number on R, and F1, F2 are fuzzy sets on V1, V2 respectively.
, is )( and is )(such that in are There: 21 FiFiIi'sQp 21 VV
3232
Fuzzy quantifiers
Any proposition p of this form can be expressed in a simplified form,
where E1, E2 are
, :' 21 'sE'sEQp
. allfor ))(()(
))(()(
2 22
1 11
IiiFiE
iFiE
V
V
3333
Fuzzy quantifiers
Moreover, p’ may be interpreted as
we may rewrite it in the form
where W is a variable taking values in R and W = | E1 ∩ E2|.
.) and ( are There :' 21 'sEEQp
, is :' Qp W
3434
Fuzzy quantifiers
Using the standard fuzzy intersection, we have
Now, for any given sets E1 and E2 ,
.)()'()( WQpTpT
,))](( )),((min[ 2 21 1
Ii
iFiF VVW
3535
Fuzzy quantifiers
• Second Kind These are quantifiers such as "almost all,"
"about half," "most," and so on. They are represented by fuzzy numbers on the unit interval [0, 1].
Examples of some quantifiers of this kind are shown in Fig. 8.5.
3636
Fuzzy quantifiers
3737
Fuzzy quantifiers
Fuzzy propositions with quantifiers of the second kind have the general form
where Q is a fuzzy number on [0, 1], and the meaning of the remaining symbols is the same as previously defined.
, is )(such that in
are there is )(such that Iin Among:
22
11
FiIQi's
Fii'sp
V
V
3838
Fuzzy quantifiers
Any proposition of the this form may be written in a simplified form,
where E1, E2 are fuzzy sets on X defined by
, are :' 21 'sQE'sQEp
. allfor ))(()(
,))(()(
2 22
1 11
IiiFiE
iFiE
V
V
3939
Fuzzy quantifiers
we may rewrite p’ in the form
where
for any given sets E1 and E2.
, is :' Qp W
.||
||
1
21
E
EE W
Ii
Ii
iF
iFiF
))((
))](( )),((min[
1 1
2 21 1
V
VVW
4040
Linguistic hedges
• Linguistic hedges Given a fuzzy predicate F on X and a modifier h
that represents a linguistic hedge H, the modified fuzzy predicate HF is determined for each x X by the equation
This means that properties of linguistic hedges can be studied by studying properties of the associated modifiers.
)).(()( xFhxHF
4141
Linguistic hedges
Every modifier h satisfies the following conditions:
ns.compositio are so then k),strong(wea are and both
if moreover, and, modifiers also are with and
with of nscompositio ,modifier another given .4
versa; viceand weak is then strong, is if .3
function; continuous a is .2
.1)1( and 0)0( .1
1
gh
gh
hgg
hh
h
hh
4242
Linguistic hedges
A convenient class of functions that satisfy these conditions is the class
where α R+ is a parameter by which individual modifiers in this class are distinguished and a [0, 1]. When α < 1, hα is a weak modifier; when α > 1, hα is a strong modifier; h1 is the identity modifier.
,)( aah
43
Inference from conditional propositions
For classical logic
Assume that the variables are related by an arbitrary relation on X × Y, not necessarily a function.
Given X = u and a relation R, we can infer that Y B, where B = { y Y |<x, y> R } (Fig. 8.7a).
Similarly, given X A, we can infer that Y B, where B = { y Y |<x, y> R, x A } (Fig. 8.7b).
44
Inference from conditional propositions
Observe that this inference may be expressed equally well in terms of characteristic functions XA, XB, XR of sets A, B, R respectively, by the equation
. allfor )] ,( ),(min[sup)( Yyyxxy RAXx
B
XXX
45
Inference from conditional propositions
46
Inference from conditional propositions
For fuzzy logic Assume that R is a fuzzy relation on X x Y, and A', B' are fuzzy sets on X and
Y, respectively. Then, if R and A' are given, we can obtain B' by the equation
which is a generalization by replacing the
, allfor )] ,( ),('min[sup)(' YyyxRxAyBXx
47
Inference from conditional propositions
characteristic functions with the corresponding membership functions.
It can also be written in the matrix form as
called the compositional rule of inference.
, '' RAB
48
Inference from conditional propositions
49
Inference from conditional propositions
Viewing proposition p as a rule and proposition q as a fact, the generalized modus ponens is expressed by the following schema:
B'
A'
BA
is :Conclusion
is :Fact
is then , is If :Rule
Y
X
YX
50
Inference from conditional propositions
Example 8.1
51
Inference from conditional propositions
52
Inference from conditional propositions
Another inference rule in fuzzy logic, which is a generalized modus tollens, is expressed by the following schema:
In this case, the compositional rule of inference has the form
A'
B'
BA
is :Conclusion
is :Fact
is then , is If :Rule
X
Y
YX
)]. ,( ),('min[sup)(' yxRyBxAYy
53
Inference from conditional propositions
Example 8.2
54
Inference from conditional propositions
The generalized hypothetical syllogism is expressed by the following schema:
X, Y, Z are variables taking values in sets X, Y, Z, respectively, and A, B,C are fuzzy sets on sets X, Y, Z, respectively.
CA
CB
BA
is then , is If :Conclusion
is then , is If :2 Rule
is then , is If :1 Rule
ZX
ZY
YX
55
Inference from conditional propositions
Given R1, R2, R3, obtained by these equations, we say that the generalized hypothetical syllogism holds if
which again expresses the compositional rule of inference. This equation may also be written in the matrix form
)]. ,( ), ,(min[sup) ,( 213 zyRyxRzxRYy
.213 RRR
56
Inference from conditional propositions
Example 8.3
57
Inference from conditional and qualified propositions
• Given a conditional and qualified fuzzy proposition p of the form
p : If X is A, then Y is B is S, (8.46)
where S is a fuzzy truth qualifier, and a fact is in the form "X is A’," we want to make an inference in the form “Y is B’."
58
Inference from conditional and qualified propositions
• The method of truth-value restrictions is based on a manipulation of linguistic truth values. It involves the following four steps.
59
Inference from conditional and qualified propositions
60
Inference from conditional and qualified propositions
• Example 8.4
61
Inference from conditional and qualified propositions
62
Inference from conditional and qualified propositions
63
Inference from conditional and qualified propositions
• Theorem 8.1 Let a fuzzy proposition of the form (8.46) be given,
where S is the identity function (i.e., S stands for true), and let a fact be given in the form "X is A'," where
for all and some x0 such that A(x0)= . Then, the inference “Y is B‘ " obtained by the method of truth-value restrictions is equal to the one obtained by the generalized modus ponens, provided that we use the same fuzzy implication in both inference methods.
)(')('sup 0)(:
xAxAaxAx
]1 ,0[a a
64
Inference from qualified propositions
• Given n quantified fuzzy propositions of the form
where Qi is either an absolute quantifier or a relative quantifier, and Wi is a variable compatible with the quantifier Qi, what can we infer from these propositions?
),( is : niii ip NQW
65
Inference from qualified propositions
• Quantifier extension principle Assume that the prospective inference is expressed in
terms of a quantified fuzzy proposition of the form
The principle states the following: if there exists a function f : Rn →R such that W = f (W1, W2, …, Wn) and Q = f (Q1, Q2, …, Qn), where the meaning of f (Q1, Q2, …, Qn) is defined by the extension principle, then we may conclude that p follows from p1, p2, …, pn.
. is : QWp
66
Inference from qualified propositions
• Intersection / product syllogism
where E, F, G are are fuzzy sets on a universal set X, Q1 and Q2 are relative quantifiers (fuzzy numbers on [0, 1]), and Q1 • Q2 is the arithmetic product of the quantifiers.
'sGFE'sp
G's'sFEp
F'sE'sp
) and ( are :
are ) and ( :
are :
21
22
11
Q
Q
67
Inference from qualified propositions
Propositions p1, p2, and p may be expressed in the form
where W1 = Prop(F / E), W2 = Prop(G / E∩F), and W = Prop(F∩G / E).
, is :
, is :
, is :
2 2 2
1 1 1
QW
QW
QW
p
p
p
68
Inference from qualified propositions
• Consequent conjunction syllogism
where E, F, G are are fuzzy sets on a universal set X, Q1 and Q2 are relative quantifiers and Q is a relative quantifier given by
'sGFE'sp
G'sE'sp
F'sE'sp
) and ( are :
are :
are :
22
11
Q
Q
Q
];) ,MIN([ )]1 ,0(MAX[ 2121 QQQQQ
69
Inference from qualified propositions
that is, Q is at least MAX(0, Q1 + Q2 - 1) and at most MIN(Q1, Q2). Here, MIN and MAX are extensions of min and max operations on real numbers to fuzzy numbers (Sec. 4.5).
7070
Exercise 6
• 6.4
• 6.8
• 6.9