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Slovak University of Technology Faculty of Material Science and Technology in Trnava. Fuzzy Systems. Introduction to Fuzzy Sets and Systems. Introduction to Fuzzy Sets and Systems. - PowerPoint PPT Presentation
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Fuzzy Systems
Introduction to Fuzzy Sets and Systems
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Introduction to Fuzzy Sets and Systems
The concept of Fuzzy Logic Fuzzy Sets and Fuzzy Systems was conceived by Zadeh, a professor at the University of California at Berkley. It is presented not as a control methodology, but as a way of processing data by allowing partial set membership rather than crisp set membership or non-membership. This approach to set theory was not applied to control systems until the 70's due to insufficient small-computer capability prior to that time. Professor Zadeh reasoned that people do not require precise, numerical information input, and yet they are capable of highly adaptive control. If feedback controllers could be programmed to accept noisy, imprecise input, they would be much more effective and perhaps easier to implement.
The word fuzzy has become common knowledge and comes up in every day conversation. This comes as quite a surprise to those of us who specialize in research into fuzzy systems. Scientific methodology requires strict logic, but one can say that not much effort goes into verification of premises and assumptions. The premises and assumptions that sciences and technology worry so little about are the same axioms in mathematics, and this probably comes about because they are not logical on the whole. At preset, this problems can only be presented through human perception and experience. If premises and assumptions are not thoroughly investigated in technical fields, there is the fear of inviting big mistakes. For example, unexpected accidents in safety systems, nonsensical conclusions in information systems, automation systems that large balance all occur when design premises are far from the actual circumstances.
Science and technology do their best to exclude subjectivity, but discovery and invention originate in right hemisphere activities that are based on subjectivity, and logicizing are no more than secondary processes for gaining the assent of others. The use of subjectivity is even more effective during the process of objectification.
Some notations of crisp set theory If A,B are sets, then A is a subset of B ( AB) if xAxB for all xA If U is an universal set, we denote by P(U) set of all subset of U, P(U)={A;AU}. P(U)
is called potential set of universal se U. If U is finite and has n elements nN, it is known that P(U) is finite and has 2n
elements. It is patent that P(U) is a Boolean algebra with respect operations union (),
intersection () and complement of sets.
Some basic (standard) operation set
A B={xU;xA or xB}={xU;xA xB} (the union of sets)
A B={xU;xA and xB}=={xU;xA xB} (the intersection of sets)
Ac={xU;xA } (the complement of the set)
A - B=A\B={xU;xA and xB}={xU;xA xB} (the different of sets).
A
AB=(A-B)(B-A)= (A\B)(B\A) (the symmetric different of sets)
AB={(x,y);xA and yB}={(x,y);xA yB} (Cartesian product of sets ).
If A,B are sets, we call relation any non empty subset R AB. If R is a relation, then notation (x,y)R is the same as xRy.
Some properties of relations
The relation R is
1. left-total: if for all x in A there exists a y in B such that xRy (this property, although sometimes also referred to as total, is different from the definition of total in the next section).
2. right-total: if for all y in B there exists an x in A such that xRy.3. symmetric, if (x,y)R(y,x)R,4. reflexive, if (x,x)R5. transitive, if [(x,y)R] [(y,z)R] [(x,z)R]6. If R is symmetric, reflexive and transitive then it is relation equivalence.7. antisymmetric: if for all x and y in B it holds that if xRy and yRx then x = y. "Greater than
or equal to" is an antisymmetric relation, because if x≥y and y≥x, then x=y.8. asymmetric: if for all x and y in A it holds that if xRy then not yRx. "Greater than" is an
asymmetric relation, because if x>y then not y>x.9. functional (also called right-definite): for all x in X, and y and z in Y it holds that if xRy and
xRz then y = z.10. funkcional is surjective: if for all y in B there exists an x in X such that xRy.11. funkcional is injective: if for all x and z in A and y in B it holds that if xRy and zRy then
x = z.12. funkcional is bijective: left-total, right-total, functional, and injective.
Mapping (function) A onto B.
If non empty relation fAB have following properties
1) for all xA there exists yB so (.x,y)f2) If [(.x,y1)f and (.x,y2)f]y1=y2.
then f is also called mapping (function) A onto B.
Notations (x,y)f, y=f(x), f:xy are equivalent.The mapping O:AA ... AA is n-ary operation.If n=2 we have binary operation.If n=1 we have u-nary operationIf O(x,y)=O(y,x) then the binary operation is commutative.If O(x, O(y,z))= O(O(x,y), z) then the binary operation is associative
Ax
AxxA ,0
,1
Characteristic function of set If A is a subset of universal set U, then function defined on U as follows
Is a characteristic function of subset A.
It is easy to show that P(U) and set of all characteristic functions CH(U) are isomorphic (as sets). There exist bijection P(U) onto CH(U) i.e. there exists two maps : P(U) CH(U) and :
CH(U) P(U) defined by (A)=A, (A)={xU; A(x)=1}=A thus CH(U) P (U).
P(U) is a Boolean algebra with respect operation union, intersection and complement. This means that following eight identities are valid
Ax
AxxA ,0
,1
Propperties of set operations 1) AB=BA, AB=BA (commutavity)
2) (AB)C=A(BC), (AB)C=A(BC) (associativity)
3) (AB)C=(AC|(BC), A (BC) = (AB)(AC) (distributivity)
4) AA=A, AA=A (idenpotency)
5) A(AB)=A, A(AB)=A (absorption)
6) A=A, A=, UU=U, UA=A
7)
8) , ,
(´)(x)=max{(x), ´(x)} (´)(x)=min {(x), ´(x)} ´(x)=1-(x)
UAA AA
AA U U
Definition of fuzzy set
Definition 1.1: Definition of fuzzy set: Let U is an universal set and . A fuzzy set is a pair {U,}. A function we call the membership function.
The value of membership function is a degree of membership of x as an element of set.
The membership function is a graphical representation of the magnitude of participation of each input. It associates a weighting with each of the inputs that are processed, define functional overlap between inputs, and ultimately determines an output response.
The rules use the input membership values as weighting factors to determine their influence on the fuzzy output sets of the final output conclusion. Once the functions are inferred, scaled, and combined, they are defuzzified into a crisp output which drives the system. There are different memberships functions associated with each input and output response.
Example Error Membership FunctionExample Membership Function Figure illustrates the features of the triangular membership function which is
used in this example because of its mathematical simplicity. Other shapes can be used but the triangular shape lends itself to this illustration.
The degree of membership is determined by plugging the selected input parameter (error or error-dot) into the horizontal axis and projecting vertically to the upper boundary of the membership function (s).
Difference between crisp set (a) and fuzzy set (b)
Some notations of fuzzy set Let is a fuzzy set. Then
a support of fuzzy set is Supp A =;
if support of fuzzy set is finite then is discreet;
-cut of fuzzy set is A =;
-level of fuzzy set is A =;
a kernel of fuzzy set is Ker A =;
if Ker A then is normal else is subnormal;
a height of fuzzy set is ;
a singleton of fuzzy set is the set with one element;
if then the fuzzy set is crisp (conventional
0; xUx A
A~
A~
A~
A~
xUx A;
xUx A;
1; xUx A
A~
A~
A~
1,0xA
Some notations on fuzzy sets
1,0xA 1,0xA
Representation theorem of fuzzy set
In applications of mathematics the useful notation is a number of elements of set (cardinality of set).
Theorem: Let 01 and A, A are cuts of fuzzy set A
~ . Then A A.
Proof: Let and A
~ is fuzzy with universe U, Then A = ; Ax U x = xUx A; xUx A; A.
(Representation theorem of fuzzy set): Let A
~ is fuzzy set
A~ = (U,A). Then A
~ = a0,1
{A ; 0,1 }.
Theorem: Let A
~ =A;0,1 . Then its membership function is
AxxA ;1,0sup)(
Example
Example: Let 0 pre ,
10 pre 25,2
A is horizontal definition of
fuzzy set. What is vertical? Solution: The graph of system
0 pre ,
10 pre 25,2
A
is on fig. For -cuts I valid
,5 pre,0
5,3 pre,2
53,2 xpre2,-x
,2- xpre,0
x
xxxA
Fuzzy Systems
Measures on Fuzzy Sets
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
The cardinality of fuzzy set In the case of finite crisp (convential) sets the number of elements set A is
Ax
AUx
A xxACardA )()(
An extension of this term on fuzzy set is
Definition: Let A
~ is fuzzy set and its universe is finite then the cardinality fuzzy set A
~
is
Card A~
= SuppAx
A x)(
It is useful to have some variable(measure) as mean membership of elements of fuzzy set. Those characteristic is Definition: Let A
~ is fuzzy set and its universe is finite then the relative cardinality
fuzzy set A~
is
card A~
=
SuppAxA
SuppAxA
x
x
)(
)(
Example: Let A
~= (0,0.5), (1,0.7), (2,0), (3,0.8), (4,1), (5,0.7), (6,0.1), (7,0), (8,0.4),
(9,0.9), (10,0.7) then Card A~
=0,5+0,7+0.8+1+0,7+0,1+0,4+0,9+0,4=
=5,5 and card A~
=SuppA
Card A~
= 5.0
11
5.5
It means that mean value of membership elements of fuzzy set A~
is 0.5. In this example is Supp A =0,1,2,4,5,6,8,9,10 . If =0.7 then -cut of A
~ is
A0.7= 1,3,4,5,9,10 and for =0.7 -level A0,7= 1,5,10 . Its kernel is Ker A= 4 . If universal set infinite, membership function is integrable and Supp A is measurable then relative cardinality of fuzzy set is defined as
Definition 5: Let membership function is integrable on measurable set Supp A. then the relative cardinality fuzzy set A
~ is
card A~
=
SuppA
SuppAA
dx
dxx)(
Example: Let
1,0,0
1,0,
x
xxxA is of membership function of fuzzy set. What is
the relative cardinality of the fuzzy set? Solution: Let us compute
card A~
= 3
2
3
2)(
1
03
1
0
1
0
x
dx
dxx
dx
dxx
SuppA
SuppAA
Center of fuzzy setWe often need the representative of object(fuzzy set). It usually is mean value or center of object. The center is easy interpreted as a representative of object (fuzzy set). Definition 6: Let membership function is integrable on measurable set Supp A. he
coordinates of center fuzzy set are
SuppAA
SuppAAi
idxx
dxxx
t)(
)(
, for i=1,2,…,n
Example: Let
2,0,0
2,0,8
3
x
xx
xA is of membership function of fuzzy set. What is
the centre of the fuzzy set ?
Solution: Let us compute
5
8
16
32.
5
4
32
140
1
8
8
)(
)(
2
04
2
05
2
0
3
2
0
3
x
x
dxx
dxx
x
dxx
dxxx
t
SuppAA
SuppAAi
i
Measure of uncertaintyFuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set, membership function and the classical probability measures. In the fuzzy measure theory, the conditions are precise, but the information about an element alone is insufficient to determine which special classes of measure should be used. The central concept of fuzzy measure theory is fuzzy measure, which was introduced by Sugeno in 1974. Fuzzy measure m: F(U)R+ can be considered as generalization of the classical probability measure. A fuzzy measure m over a set U (the universe of discourse with the subsets A~ , B~ ,...) satisfies the following conditions when U is finite: 1. m(A) 0.
2.
11 ii
ii AmAm
Definition of measure of uncertainty Let F(U) is a set of all fuzzy sets over the universe of discourse U. Then measure of uncertainty is a function m: F(U)R+ , if
1. for all A~ , B~ F(U) is m( A~ )+m( B~ )=m( A~ B~ )+m( A~ B~ ),
2. If A(x)0,1 for all xU then m( A~ )=0, U.(if fuzzy set is crisp set then measure of uncertainty is zero).
3. If A(x)=0,5 for all xU then m(B~ )m( A~ )for all B~ F(U)
4. Let A(x) B(x) if B(x)0.5 and A(x) B(x) if B(x) 0.5. Then
m( A~ )m(B~ ).
Measure of uncertainty of discrete fuzzy setHamming’s measure of uncertainty of fuzzy set is
SuppAx
AA xxAHm5.0
)(
If we compute
Ux
AA
UxAHm5.0
)(max~
Norm Hamming’s measure of uncertainty of fuzzy set is
U
)A(Hm2)A(Hm
Euclid’s measure of uncertainty Euclid’s measure of uncertainty of fuzzy set is
2Ux
5.0AA xx)A(Eu
If we compute
Ux
AA
UxAEu
2)(max
5.0~
Norm Euclid’s measure of uncertainty of fuzzy set is
U
)A(Eu2)A(Eu
Entropic measure of uncertainty of fuzzy set Entropic measure of uncertainty of fuzzy set is
UxAAAA x1lnx1xlnx)A(Ent
If we compute
Ux
AAAAA
xxxxAEnt5.05.05.05.0
1ln1ln)(max~
2ln2
11ln
2
11
2
1ln
2
1U
Ux
Norm entropic measure of uncertainty of fuzzy set is
2ln.U
)A(Ent)A(Ent
Where - U number (cardinality of U) of element of U
- xA is value of membership function in x,
- 2
15.0
xA is value of characteristic function of A0.5 cut fuzzy set A~ .
An example
Let A~ = (-1,0.5),(0,0.2),(5,1),(7.0.9),(10,0.2) . Calculate standardized Hamming's, Euclid's and entropic measure uncertainty of fuzzy set. Universal set is U= -1,0,5,7,10) and A0.5 cut of A~ is A0.5=x;A(x)0.5= -1,5,7 and
characteristic function is
5.0
5.05.0A Ax,0
Ax,1 . Then
.1
02.019.01102.015.0xx)A(HmUx
5.0AA
4.05
1.2
U
)A(Hm2)A(Hm
An example 2
.1
02.019.01102.015.0xx)A(HmUx
5.0AA
4.05
1.2
U
)A(Hm2)A(Hm
58,034,0
02.019.01102.015.0
xx)A(Eu
22222
2
Ux5.0AA
52.05
58,0.2
U
)A(Eu2)A(Eu
Ux
AAAA x1lnx1xlnx)A(Ent
65,12,01ln2,01-
9,01ln9,015,01ln5,012,01ln2,015,01ln5,01
2,0ln2,09,0ln9,01ln12,0ln2,05,0ln5,0
48,069,0*5
65,1
2ln.U
)A(Ent)A(Ent
Measure of uncertainty of fuzzy set if U=RHowever each case employed measure of uncertainty must it satisfy common condition 1-4 of definition of measure uncertainty of fuzzy set. For instance, let fuzzy set is up universal real numbers and supporter fuzzy set is interval <a,b> and let membership function on <and,b> integrable. Let
5,0
,
,11)(
x
inak
ak
x
xxxg A
A
AA
Then measure uncertainty is
b
a
dxxgAm )(
Ant its standardized (norm) version will be
ab
AmAm
)(2
Or
2 ( )b
a
M a g x dx
And its standardized (norm) version will be
4 ( )f A
M Ab a
Demonstrate, that function is measure of uncertainty fuzzy set stands to prove it satisfy conditions
1. m( A~ )+m(B~ )=m( A~ B~ )+m( A~ B~ ), 2. m( A~ )=0, if A(x)0,1 for all xU 3. Let A(x)=0,5 for all xU. Then m(B~ )m( A~ ) for all B~ F(U)
4. Let A(x) B(x) for every x, for which B(x)0.5and let A(x) B(x) for every x, for which B(x) 0.5 Then m( A~ )m( B~ )
The condition 1 of definition measure of uncertainty pays because integral is contents
of plane areas, it is measure and every measure it must satisfy. Count BAmBAmBABmBAmBAAmBAm
BmAm )()(
)(\)(\)(
BAmBmAm )()( and so
)( BAm BAmBmAm )()(
BAmBAm )( )()( BmAm
BAmBAm )( )()( BmAm
Fuzzy complement. Membership functionA fuzzy set operation is an operation on fuzzy set. These operations are generalization of crisp set operations. There is more than one possible generalization. The most widely used operations are called standard fuzzy set operations or elementary operations. There are three operations: fuzzy complements, fuzzy intersections and fuzzy union. Standard fuzzy complement is defined by
µc(x)=1- µ(x), for all xU
Membership function of fuzzy intersection Standard fuzzy intersection is defined by
µAB (x)=min{ µA(x), µB(x)} for all xU
Membership function of fuzzy unionStandard fuzzy union is defined by
µAB (x)= max { µA(x), µB(x)} for all xU
Fuzzy complementsMembership function µ(x) is defined as the degree to which x belongs to. Let c A~
denote a fuzzy complement of A~ of type c. Then cµ(x) is the degree to which x belongs to c A~ , and the degree to which x does not belong to A~ . (µ(x) is therefore the degree to which x does not belong to c A~ .)
Axioms for fuzzy complements
The generalization of standard complements of fuzzy set is any operation satisfying next axioms
Axiom c1. Boundary condition C(0) = 1 and C(1) = 0
Axiom c2. Monotonicity
For all a, b <0, 1>, if a ≤ b, then C(a) ≥ C(b) Axiom c3. Continuity
C is continuous function. Axiom c4. Involutions
C is an involution, which means that C(C (a)) = a for each a <0,1>. Standard fuzzy complement satisfy axioms c1-c4. We prove it
Axiom c1: If c(µ(x))= µc(x) =1-µ(x) then µc(0) =1-0=1 and µc(1) =1-1=0 Axiom c2: If c(µ(x))= µc(x) =1-µ(x) and a≤b then –a≥-b and 1-a≥1-b Axiom c3: If c(µ(x))= µc(x) =1-µ(x) then y=1-t is elementary and so continue function. Axiom c4: If c(µ(x))= µc(x) =1-µ(x) then (µc)c(x)=1-(1-µ(x))=µ(x)
Fuzzy intersectionsThe intersection of two fuzzy sets A and B is specified in general by a binary
operation on the unit interval, a function of the form µ:[0,1]×[0,1] → [0,1]. µAB(x) =µ* [A(x), B(x)] for all x.
Axioms for fuzzy intersection
Axiom i1. Boundary condition µ* (a, 1) = a
Axiom i2. Monotonicity
b ≤ d implies µ* (a, b) ≤ µ* (a, d) Axiom i3. Commutability
µ* (a, b) = µ* (b, a) Axiom i4. Associativity
µ* (a, µ* (b, d)) = µ* (µ* (a, b), d) Axiom i5. Continuity
iµ*is a continuous function Axiom i6. Subidempotency
µ* (a, a) ≤ a
Fuzzy intersections. An example
An example of fuzzy fuzzy intersection is µ*(= µAB(x)= min {µA(x),µB(x)} To prove that we show that it satisfy axioms i1-i6. Axiom i1.: µ* (a, 1) = a. Let us compute µ* (a, 1)= µ*(= µAU(x)= =min {µA(x),1 }= µA(x), Axiom i2. If a=µA(x), µB(x)=b ≤ d= µC(x) implies µ* (a, b) ≤ µ* (a, d). Let us compute µ* (a, b)= µ*(= µAB(x)= min {µA(x), µB(x) }≤ min {µA(x),
µC(x) } Axiom i3: Commutativity µ* (a, b) = µ* (b, a). Let us compute µ* (a, b) = =min {µA(x), µB(x) }= min {µB(x), µA(x) }=µ* (b, a) Axiom i4. Associativity µ* (a, µ* (b, d)) = min {µA(x), min {µB(x), µC(x) } }= =min{min{ µA(x), µB(x)}, µC(x) }= µ* (µ* (a, b), d) Axiom i5. Continuity: µ*(= µAB(x)= min {µA(x), µB(x)}=min{u,v} is continues function Axiom i6. Subidempotency µ* (a, a) = min {µA(x), µA(x) }= µA(x)≤ µA(x)= a
Fuzzy unionsThe union of two fuzzy sets A and B is specified in general by a binary operation on
the unit interval function of the form u:[0,1]×[0,1] → [0,1]. (A B)(x) = u[A(x), B(x)] for all x
Axioms for fuzzy union
Axiom u1. Boundary condition u(a, 0) = a
Axiom u2. Monotonicity
b ≤ d implies u(a, b) ≤ u(a, d) Axiom u3. Commutativity
u(a, b) = u(b, a) Axiom u4. Associativity
u(a, u(b, d)) = u(u(a, b), d) Axiom u5. Continuity
u is a continuous function
An example of fuzzy union operation An example of fuzzy union operation is
µ*( )~
,~
BA = µAB(x)= max{µA(x),µB(x)}
To prove that we show that it satisfy axioms u1-u5.
Axiom u1. Boundary condition
u(a, 0)=µ*( )~
,~
AA = µAA(x)= max{µA(x),µA(x)}= µA (x)= a
Axiom u2. Monotonicity
b ≤ d implies u(a, b) ≤ u(a, d) Let a= µA (x),b= µB(x),d= µC(x). Then
u(a, b)=max{ µA(x),µB(x)}≤ max{ µA(x),µC(x)}=u(a,db) Axiom u3. Commutativity
u(a, b) = max{ µA(x),µB(x)}= max{ µB(x),µA(x)}=u(b, a) Axiom u4. Associativity
u(a, u(b, d)) = u(u(a, b), d)
u(a, u(b, d)) = max{ µA(x), max{ µB(x),µC(x)}}= max{ max{ µA(x),µB(x)}, µC(x }= =u(u(a, b), d) Axiom u5. Continuity
u is a continuous function
Aggregation operationsAggregation operations on fuzzy sets are operations by which several fuzzy sets
are combined in a desirable way to produce a single fuzzy set. Aggregation operation on n fuzzy set (n≥2) is defined by a function
v:<0,1>n → <0,1>
Axioms for aggregation operations fuzzy sets
Axiom v1. Boundary condition v(0, 0, ..., 0) = 0 and v(1, 1, ..., 1) = 1
Axiom v2. Monotonicity
For any pair (a1, a2, ..., an) and <b1, b2, ..., bn> of n-tuples such that ai, bi <0,1> for all i N, if ai ≤ bi for all i N, then v(a1, a2, ...,an) ≤ v(b1, b2, ..., bn); that v is monotonic increasing in all its arguments.
Axiom v3. Continuity
V is a continuous function.
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operations used in the framework of probabilistic spaces and in multi valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary metric spaces.
Definition A t-norm is a function T: <0, 1> × <0, 1> → <0, >] which satisfies the following properties:
Commutavity: T(a, b) = T(b, a) Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d Associativity: T(a, T(b, c)) = T(T(a, b), c) The number 1 acts as identity element: T(a, 1) = a
Since a t-norm is a binary algebraic operation on the interval <0, 1>, infix algebraic notation is also common, with the t-norm usually denoted by * .
The defining conditions of the t-norm are exactly those of the partially ordered Abelian monoid on the real unit interval <0, 1> (ordered group). The monoidal operation of any partially ordered Abelian monoid L is therefore by some authors called a triangular norm on L.
Motivations and applicationsof T-norm
T-norms are a generalization of the usual two-valued logic conjunction, studied by classical logic, for fuzzy logic. Indeed, the classical Boolean conjunction is both commutative and associative. The Monotonicity property ensures that the truth value of conjunction does not decrease if the truth values of conjuncts increase. The requirement that 1 be an identity element corresponds to the interpretation of 1 as true (and consequently 0 as false). Continuity, which is often required from fuzzy conjunction as well, expresses the idea that, roughly speaking, very small changes in truth values of conjuncts should not macroscopically affect the truth value of their conjunction.
T-norms are also used to construct the intersection of fuzzy sets or as a basis for aggregation operators.
Fuzzy Systems
Classification of t-normsand conorms
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Classification of t-normsProminent examples
Minimum t-norm yxyxT ,min,min also called the Gōdel t-norm, as it is the standard semantics for conjunction in Gōdel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the point wise largest t-norm. By the minimum t-norm Zadeh defined intersection on fuzzy sets and in many papers is called standard operation of intersection fuzzy sets.
Product t-norm:
yxyxTprod .),(
(product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm.
Łukasiewicz t-norm
1,0max), yxyxTLuk .
The name comes from the fact that the t-norm is the standard semantics for strong conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, point wise smaller than the product t-norm.
Drastic t-norm
otherwice 0,
1y if x,
1 xif ,
,
y
yxTD
The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm. It is a right-continuous Archimedean t-norm.
Nilpotent minimum
othrwice 0,
1y xif ,,min,
yxyxTnM
is a standard example of a t-norm which is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm.
Hamacher product
otherwiceyxTH ,
xy-yx
xy0y xif ,0
,0
is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer Sklar t-norms.
Properties of t-norms
The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm:
yxTyxTyxTD ,,, min
for any t-norm and all a, b in [0, 1].
For every t-norm T, the number 0 acts as null element: T(x, 0) = 0 for all x in <0, 1>.
A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval <0, x> or <0, x), for some x in <0, 1>.
Properties of continuous t-norms
Although real functions of two variables can be continuous in each variable without being continuous on <0, 1>2, this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions fy(x) = T(x, y) are continuous for each y in <0, 1>. Analogous theorems hold for left- and right-continuity of a t-norm.
A continuous t-norm is Archimedean if and only if 0 and 1 are its only idenpotents.
A continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that
yfxfTfyxT Luk ,, 1
For each continuous t-norm, the set of its idempotents is a closed subset of <0, 1>. Its complement — the set of all elements which are not idempotent — is therefore a union of countable many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm.
T-conormsT-conorms (also called S-norms) are dual to t-norms under the order-reversing
operation which assigns 1 – x to x on [0, 1]. Given a t-norm, the complementary conorm is defined by
)1,1(1, yxTyxS
This generalizes De Morgan’s laws..
It follows that a t-conorm satisfies the following conditions, which can be used for an equivalent axiomatic definition of t-conorms independently of t-norms:
Commutativity: S(a, b) = S(b, a) Monotonicity: S(a, b) ≤ S(c, d) if a ≤ c and b ≤ d Associativity: S(a, S(b, c)) = S(S(a, b), c) Identity element: S(a, 0) = a
T-conorms are used to represent logica disjunction in fuzzy logic and union in fuzzy set theory..
Examples of t-conorms
Maximum t-conorm
yxyxS ,max,max
dual to the minimum t-norm, is the smallest t-conorm. It is the standard semantics for disjunction in Gödel fuzzy logic and for weak disjunction in all t-norm based fuzzy logics.
Probabilistic sum and bounded sum Probabilistic sum
yxyxyxS sum .,
is dual to the product t-norm. In probability theory it expresses the probability of the union of independent events.. It is also the standard semantics for strong disjunction in such extensions of product fuyyz logic in which it is definable (e.g., those containing involutive negation).
Bounded sum
1,min, yxyxSLuk
is dual to the Łukasiewicz t-norm. It is the standard semantics for strong disjunction in Lukasiewicz fuzzy logic.
Drastic t-conorm and Nilpotent maximum
Drastic t-conorm
otherwice 1,
0y if ,
0 xif ,
, x
y
yxSD
dual to the drastic t-norm, is the largest t-conorm . The function is discontinuous at the lines 1 > x = 0 and 1 > y = 0.
Nilpotent maximum
otherwice 1,
1y xif ,,max,
yxyxSnM
dual to the nilpotent minimum:
The function is discontinuous at the line 0 < x = 1 – y < 1.
Einstein sum (compare the velocity-addition formula under special relativity)
xy
yxyxSH
1,
2
is a dual to one of the Hamacher t-norms.
Properties of t-conorms
Many properties of t-conorms can be obtained by dualizing the properties of t-norms, for example:
For any t-conorm, the number 1 is an annihilating element: S(a, 1) = 1, for any a in <0, 1>.
Dually to t-norms, all t-conorms are bounded by the maximum and the drastic t-conorm:
yxSyxSyxS D ,,,max
for any t-conorm S and all x,z in <0, 1>.
Unary operations in fuzzy sets
Operation decreasing vagueness, according to definition of measure uncertainty fuzzy set ( let A(x) B(x) if B(x)0.5 and A(x) B(x) if B(x) 0.5 then m( A~ )m(B~ ), needs increase value of membership function when its value is less than 0,5 and value decrease its value if it is smaller 0,5. So we have The operation decreasing contrast is any mapping Int:F(U) F(U),
F(U)= 1,0:; UAA , if it satisfy
5.0 if and 5.0 if xxxIntxxxInt AAAAA Example: Mapping
5.0 if,2
5.0 if ,2
1
2
12)(
2 xx
xxxInt
AA
AA
Is function (operation) decreasing contrast. It is evident, because
xxxx AAAA 21122
1
2
12
If xt A2 , then
02
112011222112 22
tttttt
If t<0.5,1>, then 02
1t and t-1≤0 and 5.0 if , xxxInt AA .
If t<0,0.5>, then 2t2≤t2t≤1 t≤0.5 is true
Graph membership function of fuzzy set and Int function Function defined by
5.0 if,2
5.0 ,32)(
2
23
xx
xifxxxInt
AA
AAA
is function (operation) c too. To prove it we denote t=A(x), then
5.0 t,2
5.0 tif ,32)(
2
23
ift
ttxInt
If Int is function decreasing contrast, then if t≥0.5 it must satisfy -2t3+3t2t -2t(t-1)(t-0.5)0 t(1-t)(t-0.5)0 t-0.50 t0.5
is true. If 0≤ t≤0.5 it must satisfy
2t2≤t2t2-t≤0 t(2t-1)≤02t-1≤0 t≤0.5 is true too. So Int is function decreasing contrast.
For extension, restriction, value membership functionl we have operation concentration and dilatation.
Let Con, Dil F(U). Then Con is operation concentration if and only i )x()x(CON A
for all xU.
Dil is operation dilatation if and only i
)x()x(DIL A
for all xU. Example: Functions
1,)()( axxCON aA
10,)()( axxDIL aA
for all xU are operations concentration and dilatation. For instance
2A )x()x(CON
)x()x(DIL A
Graphs functions Dil and Con
Fuzzy Systems
Fuzzy relations
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Fuzzy relations
Definition: (classical n-ary relation) Let X1,...,Xn be classical(crisp) sets. The subsets of the Cartesian product X1 ×···× Xn are called n-ary relation. If X1 = ··· = Xn and R Un then R is called an n-ary relation (operation) in U.
Let R be a binary relation in R. Then the characteristic function of R is defined as
Ryx
RyxyxR ),(,0
),(,1,
Example: Consider the following relation dcybaxRyx ,,,
dcbayx
dcbayxyxR ,,),(,0
,,),(,1,
Let R be a binary relation in a classical set X. Then
Graph relation R
Properties of relationsDefinition. (reflexivity) R is reflexive if (x,x) R for all xU.
Definition. (anti-reflexivity) R is anti-reflexive if f (x,x) R for all xU.
Definition. (symmetricity) R is symmetric if from (x,y) R (y,x) R for all x,yU.
Definition. (anti-symmetricity) R is anti-symmetric if (x,y) R and (y,x) R then x=y for all x,yU.
Definition. (transitivity) R is transitive if (x, y) R and (y,z)R R then (x, z) R, for all x,y,zU.
Example. Consider the classical inequality relations on the real line R. It is clear that ≤ is reflexive, anti-symmetric and transitive, < is anti-reflexive, antisymmetric and transitive.
Other binary relations are
Definition. (equivalence) R is an equivalence relation if R is reflexive, symmetric and transitive
Example.
The relation = on natural numbers is equivalence relation.
Definition. (partial order) R is a partial order relation if it is reflexive, antsymmetric and transitive.
Definition. (total order) R is a total order relation if it is partial order and for all x,yU (x,y)R or (y,x)R.
Example. Let us consider the binary relation ”subset of”. It is clear that we have a partial order relation.
The relation ≤ on natural numbers is a total order relation.
Fuzzy relationLet U and V be nonempty sets. A fuzzy relation R is a fuzzy subset of U × V .
In other words, R F (U × V ), 1,0: VUR
It is often used equivalence notation ),(),( yxRyxR .
If U =V then we say that R is a binary fuzzy relation in U.
Let R be a binary fuzzy relation on R. Then R(x,y) is interpreted as the degree of membership of the ordered pair (x,y) in R.
Example. A simple example of a binary fuzzy relation on U = {1, 2, 3},
called ”approximately equal” can be defined as R(1, 1) = R(2, 2) = R(3, 3)=1,R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2)=0.8 , R(1, 3) = R(3, 1)=0.3
The example of fuzzy relation
The example. A simple example of a binary fuzzy relation on U = {1, 2, 3},
called “approximately equal” can be defined as R(1, 1) = R(2, 2) = R(3, 3)=1,R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2)=0.8 , R(1, 3) = R(3, 1)=0.3
In matrix notation it can be represented as
18.03.0
8.018.0
3.08.01
Operations on fuzzy relations The intersection
Fuzzy relations are very important because they can describe interactions between variables. Let R and S be two binary fuzzy relations on X × Y .
Definition: The intersection of R and S is defined by
(R S)(x,y) = min{R(x,y),S(x,y)}.
Note that R : U ×V → <0, 1>, i.e. R the domain of R is the whole Cartesian product U × V .
Definition: The union of R and S is defined by
(R S)(x,v) = max{R(x, z),S(x, z)} Example: Let us define two binary relations
R = ”x is considerable larger than y”=
8.0
0
7.0
7.019.0
08.00
1.01.08.04
3
2
1
321 y
x
x
x
yyy
5.0
0
6.0
7.003.0
04.00
1.004.04
3
2
1
321 y
x
x
x
yyy
S = ”x is very close to y”=
The intersection of R and S means that ”x is considerable larger than y” and
„is very close to y”.
(R S)(x,y) =min{R(x,y),S(x,y)}=
The union of R and S means that ”x is considerable larger than y” or ”x is very close to y”.
8.0
7.0
7.0
8.019.0
5.08.09.0
9.008.04
3
2
1
321 y
x
x
x
yyy
The basic properties of fuzzy relationsWe wil now try to give some basic properties of compositions of fuzzy relations
which plays a major role in areas such as fuzzy control, fuzzy diagnosis and fuzzy
expert systems.
1. RRIIR
2. OROOR
3. In general RSSR
4. RRRR mm 1
5. mnnm RRR
6. mnnm RR
7. )()( TSRTSR
8. TRSRTSR )(
9. TRSRTSR )(
10. TRSRTS
Fort inverse relarions
11. ccc SRSR ccc SRSR ccc SRSR
12. RRcc
13. cc SRSR
Let R* I fuzzy equivalence relation and R*(x,y)≥R(x,y) and for any fuzzy
equivalence relation S, S(x,y)≥R*(x,y), then R* is minimum fuzzy equivalence closer of
R.
Example: Let
What is minimum fuzzy equivalence closer of R?
The minimum fuzzy equivalence closer of R is fuzzy reflexive relation. The fuzzy
relation is reflexive if for all xU R(x,x)=1. The minimum reflexive relation R*R is relation
R*(x,x)=1 and R*(x,y) =R(x,y) for all xy. Hence
The fuzzy relation is symmetric if for all x,yU R(x,y)=R(y,x). The minimum symmetric relation R*R is relation R*(x,y)=max {R(x,y),R(z,x)} for all xy. Hence
8.04.02.06.0
7.04.05.02.0
7.05.013.0
7.5.02.09.0
R
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
*R
H e n c e
T h e m i n i m u m f u z z y t r a n s i t i v e r e la t i o n f u z z y c lo s e r o f R a n d i f U i s f i n i t e t h e n
R * = R n - 1 . H e n c e
7.07.07.0
7.015.05.0
7.05.013.0
7.05.03.01
17.0,4.0max7.0,2.0max7.0,6.0max
7.0,4.0max15.0,5.0max5.0,2.0max
7.0,2.0max5.0,2.0max13.0,2.0max
7.0,6.0max5.0,2.0max3.0,2.0max1
*R
17.07.07.0
7.015.05.0
7.05.013.0
7.05.03.01
17.07.07.0
7.015.05.0
7.05.013.0
7.05.03.01
2 R
1,7,.7,.7.max7,.7,.5,.5.max7,.5,.7,.3.max7,.5,.3,.7.max
7,.7,.5,.5.max7.1,5,.5.max7,.5,.5,.3.max7,.5,.3,.5.max
7,.5,.7,.3.max7,.5,.5,.3.max7,.5,.1,3.max7,.5,.3,.3.max
7,.5,.3,.7.max7,.5,.3,.5.max7,.5,.3,.3.max7,.5,.3,.1max
17.07.07.0
7.017.07.0
7.07.017.0
7.07.07.01
17.07.07.0
7.015.05.0
7.05.013.0
7.05.03.01
17.07.07.0
7.017.07.0
7.07.017.0
7.07.07.01
23 RRR
1,7,.7,.7.max7,.7,.7,.7.max7,.7,.7,.7.max7,.7,.7,.7.max
7,.7,.7,.7.max7.1,5,.5.max7,.7,.5,.5.max7,.5,.3,.5.max
7,.7,.7,.7.max7,.7,.5,.5.max7,.5,.1,3.max7,.5,.3,.3.max
7,.7,.7,.7.max7,.5,.3,.5.max7,.5,.3,.3.max7,.5,.3,.1max
17.07.07.0
7.017.07.0
7.07.017.0
7.07.07.01
I f f u z z y r e l a t i o n s i s n o t s y m m e t r i c t h e n f o r s y m m e t r i c c l o s e r o f R p a y
R * ( x , y ) ? R ( x , y ) a n d R * ( x , y ) ? R ( y , x ) . A t f i r s t w e t a k e R * ( x , y ) = m a x { R ( y , x ) , R ( x , y ) } . I t c a n
b e i n t e r e s t i n g t o t a k e R * ( x , y ) = m i n { R ( y , x ) , R ( x , y ) } .
E x a m p l e : L e t
T h e n t h e f i r s t e s t i m a t i o n o f R * i s
T h e m i n i m u m f u z z y t r a n s i t i v e r e l a t i o n f u z z y c l o s e r o f R ´ , f U i s f i n i t e , i s R * = R n - 1 .
H e n c e
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
R
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
´R
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
2 R
If fuzzy relations is not symmetric then for symmetric closer of R pay
R*(x,y)≥R(x,y) and R*(x,y)≥R(y,x). At first we take R*(x,y)=max{ R(y,x), R(x,y) }. It can
be interesting to take R*(x,y)=min{ R(y,x), R(x,y) }.
Example: Let
Then the first estimation of R* is
The minimum fuzzy transitive relation fuzzy closer of R´, f U is finite, is R*=Rn-1.
Hence
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
R
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
´R
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
2 R
Projections on axis Consider a classical relation R.
dcbayx
dcbayxyxR
,,),(,0
,,),(,1,
It is clear that projection (or shadow) of R on the X-axis is the closed interval <a, b> and its projection on the Y -axis is <c,d>.
Definition: If R is a classical relation in U × V then
ΠX = {x U| y V :(x, y) R}
ΠY = {yV |x U :(x, y) R}
Where ΠX denotes projection on U and
ΠY denotes projection on V.
Definition: Let R be a fuzzy binary fuzzy relation on U × V . The projection of R on U is defined as
ΠX(x) = sup{R(x, y) | y V }
and the projection of R on Y is defined as
ΠY (y) = sup{R(x, y) | x U}
Example: Consider the relation
R = ”x is considerable larger than y”=
8.0
0
7.0
7.019.0
08.00
1.01.08.04
3
2
1
321 y
x
x
x
yyy
then the projection on X means that
•x1 is assigned the highest membership degree from the tuples (x1,y1), (x1,y2), (x1,y3), (x1,y4), i.e. ΠX(x1)=1, which is the maximum of the first row.
•x2 is assigned the highest membership degree from the tuples (x2,y1), (x2,y2), (x2,y3), (x2,y4), i.e. ΠX(x2)=0.8, which is the maximum of the second row.
•x3 is assigned the highest membership degree from the tuples (x3,y1), (x3,y2), (x3,y3), (x3,y4), i.e. ΠX(x3)=1, which is the maximum of the third row.
Shadows of a fuzzy relation Definition: The membership function of Cartesian product of A
~F (U) and
B~F (V) is defined as
( A~
× B~
)(x,y) = min{A(x),B(y)}.
for all xU and yV.
Fuzzy Systems
Cartesian product of fuzzy sets
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Cartesian product of fuzzy setsIt is clear that the Cartesian product of two fuzzy sets is a fuzzy relation.
If A and B are normal then ΠY (A × B)= B and ΠX(A × B)= A.
Really,
ΠX(x) = sup{(A × B)(x, y) | y}
= sup{A(x) ∧ B(y) | y} = min{A(x), sup{B(y)}| y}
= min{A(x), 1} = A(x).
Definition: The sup-min composition of a fuzzy set C~F (U) and a fuzzy relation R F
(U × V ) is defined as
(C~ R)(y) =
Uxsup {min{C(x),R(x, y)}}
for all yV .
The composition of a fuzzy set C~
and a fuzzy relation R can be considered as the
shadow of the relation R on the fuzzy setC~
.
Cartesian product of fuzzy sets.
Example 1Let R = A
~ × B
~ Is fuzzy relation.
Observe the following property of composition A~ R = A
~ ( A~
× B~
)= A~
,
B~ R = B
~ ( A~
× B~
)= B~
.
Example: Let C~
be a fuzzy set in the universe of discourse {1, 2, 3} and let R be a binary fuzzy relation in {1, 2, 3}. Assume that
C~
={(1,0.2),(2,1)(3,0.3)} and R=
18,03,0
8.018.0
3.08.01
Using the definition of sup-min composition we get
C~ R=(0.2,1,0.3)
18,03,0
8.018.0
3.08.01
=(max{min{0.2,1},min{1,0.8},min{0.3,0.3}},
max{min{0.2,0.8},min{1,1},min{0.3,0.8}},max{min{0.2,0.3},min{1,0.8},min{0.3,1}}= =(0.8,1,0.8).
Example 2Example: Consider two fuzzy relations
R = ”x is considerable larger than y”=
S = ”y is very close to z” =
Then their composition is
RS=
7.09.07.0
04.00
5.08.06.0
0.50.76.0
0.80.59.0
00.4 0
0.30.94.0
4
3
2
1
321
y
y
y
y
zzz
0.50.76.0
0.80.59.0
00.4 0
0.30.94.0
8.0
0
7.0
7.019.0
08.00
1.01.08.0
4
3
2
1
3214
3
2
1
321
y
y
y
y
zzzy
x
x
x
yyy
5.0,7.0,0,3.0max7.0,5.0,4.0,9.0max6.0,7.0,0,4.0max
0,0,0,0max 0,0,4.0,0max 0,0,0,0max
5.0,1.0,0,3.0max7.0,1.0,1.0,8.0max6.0,1.0,0,4.0max
sup-product composition of fuzzy relationsDefinition: (sup-product composition of fuzzy relations) Let R F (U × V ) and S F (V × T). The sup-product composition of R and S, denoted by RS is defined as
(R S)(x,z) = zySyxRVy
,.,sup
It is clear that R S is a binary fuzzy relation in U×T.
Example: Consider two fuzzy relations
R = ”x is considerable larger than y”=
S = ”y is very close to z” =
Then their sup-product composition is
RS=
=
0.50.76.0
0.80.59.0
00.4 0
0.30.94.0
4
3
2
1
321
y
y
y
y
zzz
0.50.76.0
0.80.59.0
00.4 0
0.30.94.0
8.0
0
7.0
7.019.0
08.00
1.01.08.0
4
3
2
1
3214
3
2
1
321
y
y
y
y
zzzy
x
x
x
yyy
4.0,56.0,0,27.0max56.0,35.0,4.0,81.0max48.0,63.0,0,36.0max
0,0,0,0max 0,0,72.0,0max 0,0,0,0max
35.0,08.0,0,24.0max49.0,5.0,04.0,72.0max42.0,09.0,0,32.0max
56.081.063.0
072.00
35.072.042.0
If possible to define composition of fuzzy relations in another manner. For instance, operator max we can replace any t-conorm and min any t-norm. Fuzzy relation is Reflexive if R(x,x)=1 for all xU. Symmetric if R(x,y)=R(y,x) for all (x,y)R Transitive if Total if for all xU R(x,y) >0 or R(y,x)>0. Anti symmetric if R(x,y) >0 and R(y,x)>0 implies x=z. Strongly fuzzy transitive if
for all (x,y)R
It is clear there exist a fuzzy transitive relations R* that R* is strongly
transitive and R*(x,y)≥R(x,y)(for example R*(x,y)=1).
Let R* is strongly transitive relations and R*(x,y)≥R(x,y) and for any
strongly transitive transitive relation S,S(x,y)≥R(x,y) S(x,y)≥R*(x,y), then R* is
fuzzy transitive closer of R.
If U is reflexive and has n elements, then
1
1 ...n
RRRRn is transitive
closer of R.
),().,(supy)R(x, yzRzxRUz
Example Let
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
2 R
1,4,.2,.6.max4,.4,.2,.5.max2,.4,.2.2.max6,.2,.2,.6.max
7,.7,.5,.2.max4.1,5,.2.max4,.5,.5,.2.max6,.2,.3,.2.max
7,.5,.7,.3.max4,.5,.5,.3.max2,.5,.1,2.max6,.2,.3,.3.max
7,.5,.2,.7.max4,.5,.2,.5.max2,.5,.2,.2.max6,.2,.2,.1max
15.04.06.0
7.015.06.0
7.05.016.0
7.05.05.01
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
15.04.06.0
7.015.06.0
7.05.016.0
7.05.05.01
23 RRR
1,5,.4,.6.max4,.5,.4,.5.max2,.5,.4.2.max6,.2,.3,.6.max
7,.7,.5,.6.max4.1,5,.5.max2,.5,.5,.2.max6,.2,.3,.6.max
7,.5,.7,.6.max4,.5,.5,.5.max2,.5,.1,2.max6,.2,.3,.6.max
7,.5,.5,.7.max4,.5,.5,.5.max2,.5,.5,.2.max6,.2,.3,.1max
15.05.06.0
7.015.06.0
7.05.016.0
7.05.05.01
Example: The relation
107.0
015.0
7.05.01
R is reflexive(R(x,x)=1 for all x) and
symmetric(R(1,2)=R(2,1)=0.5, R(1,3)=R(3,1)=0.7, R(2,3)=R(3,2)=0) and so is is fuzzy
similarity reletion.
The converse fuzzy relation is usually denoted as Rc is defined as
Rc (x,y)=R(y,x)
For all x,yU
Identity relation
I(x,x)=1 for all xU
I(x,y)=0 for all xyU
Zero relation
o(x,y)=0 for all x,yU
Universe relation
Example: The following are examples of these relations
107.0
015.0
1.02.01
101.0
012.0
7.05.01cRR
107.0
015.0
7.05.01
R
000
000
000
O
111
111
111
U
Let R* is reflexive, symmetric and is strongly fuzzy transitive relation
then R* is fuzzy similarity relation often called fuzzy equivalence relation.
If fuzzy relations is not symmetric then for symmetric closer of R pay
R*(x,y)≥R(x,y) and R*(x,y)≥R(y,x). At first we take R*(x,y)=max{ R(y,x), R(x,y) }.
It can be interesting to take R*(x,y)=min{ R(y,x), R(x,y) }.
Example: Let
Then the first estimation of R* is
14.02.06.0
7.015.02.0
7.05.013.0
7.5.02.01
R
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
´R
The minimum fuzzy transitive relation fuzzy closer of R´, f U is finite, is R*=Rn-1.
Hence
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
2 R
1,4,.2,.6.max4,.4,.2,.2.max4,.4,.2,.2.max6,.2,.2,.6.max
4,.4,.2,.2.max4.1,5,.2.max2,.5,.5,.2.max4,.2,.2,.2.max
4,.4,.2,.2.max2,.5,.5,.2.max2,.5,.1,2.max2,.2,.2,.2.max
6,.2,.2,.6.max4,.2,.2,.2.max2,.2,.2,.2.max6,.2,.2,.1max
14.04.06.0
4.015.04.0
4.05.012.0
6.04.02.01
14.02.06.0
4.015.02.0
2.05.012.0
6.02.02.01
14.04.06.0
4.015.04.0
4.05.012.0
6.04.02.01
3 R
14.04.06.0
4.015.04.0
4.05.012.0
6.04.02.01
T-indistinguishability relationDefinition. T-indistinguishability relation E is a reflexive and symmetric fuzzy relation such that T(E(x,y),E(y,z))≤E(x,z) for all x,y,zU.
Definition. A S-pseudometric m is a mapping m:UU <0,1> such that -m(x,x)=0 -m(x,y)=m(y,x) S(m(x,y),m(y,z))≥m(x,z)
for all x,y,zU. There is a close relation between T-indistinguishability relations and
S- pseudometrics as is shown in the following theorem: Theorem. Let E be a T- indistinguishability relation and let be a continous order-reversing bijection from <0,1> to <0,1>. Then
mE(x,y)=(E(x,y)) is a S-pseudometric.
To be more concrete, in order to apply the transitive closure method to construct a similarity relation and, in general, a fuzzy T-transitive relation, a reflexive and symmetric fuzzy relation has to be used as a starting point. In others words, an index of similarity relating each couple of elements in the sample set has to be given: each two elements should be matched, in some way, and then the method is applied to obtain either a similarity or dissimilarity measure. At this point, the first arising question is the following: Does it mean that, for instance, from a single criterion, or from the matching of all elements to one given, no similarity measure can be given? The obvious negative answer can be stated by assuming that as a result of the single criterion evaluation or the matching-to-one process, a function
h:U <0,1>
is given, h(x) representing the degree to which x fits the given conditions. In this assumption it is easy to check that
m(x,y)=h(x)-h(y)
is a pseudo-distance on U. It is also quite obvious, that
E(x,y)=1-m(x,y)
is a likeness relation on U . It is the measure of similarity between the element y , and any perfect prototype.
For a long time, the only available methods to build up fuzzy transitive relations have been the transitive closure and related methods. As it has been pointed out repeteadly, these methods carry on a number of major problems, like the requirements of both storage and computer-time and, in spite of this, no one is satisfied with the results they yield, because there is no way to control the distorsion that its application produces on the data sample, so that the transitive closure methods do not fit the desiderata of having a method to specify a similarity measure which matches with the data.
To be more concrete, in order to apply the transitive closure method to construct a similarity relation and, in general, a fuzzy T-transitive relation, a reflexive and symmetric fuzzy relation has to be used as a starting point. In others words, an index of similarity relating each couple of elements in the sample set has to be given: each two elements should be matched, in some way, and then the method is applied to obtain either a similarity or dissimilarity measure. At this point, the first arising question is the following: Does it mean that, for instance, from a single criterion, or from the matching of all elements to one given, no similarity measure can be given? The obvious negative answer can be stated by assuming that as a result of the single criterion evaluation or the matching-to-one process, a function
h:U <0,1>
is given, h(x) representing the degree to which x fits the given conditions. In this assumption it is easy to check that
m(x,y)=h(x)-h(y)
is a pseudo-distance on U. It is also quite obvious, that
E(x,y)=1-m(x,y)
is a likeness relation on U . It is the measure of similarity between the element y , and any perfect prototype.
Such a construction can be extended in order to get T-transitive fuzzy relations for any t-norm. If T* stands for the quasi-inverse of the t-norm T , i.e. then it is also easy to check that
yhxhyhxhTyxE ,min,max),( *
is a T-fuzzy transitive relation, such that
),()( 0xxExh
for any 110
hx . Thus, for instance,
yhxh
yhxhyhxhyxE
,1
,,min),(
is a the similarity relation induced by h , i.e. E is min-transitive. On its own part, yhxh
yhxhyxE
,max
,min),(
is a probabilistic relation, i.e. transitive with respect to the t-norm T(a,b)=a.b and m(x,y)=1-E(x,y)
is a generalized pseudo-metric with respect to the t-conorm s(a,b)=a+b-a.b. Summing up, the above considerations show what to do in order to obtain a similarity measure which matches to the data from a single symmetrica evaluation of the degrees of similarity in the sample set. Next, suppose that several criteria or prototypes are given in the form of a family of functions 1,0: Uh j in this case the most natural procedure
seems, first, to get the similarity measure –in the form of a fuzzy transitive relation for a fixed t-norm T – associated with each hj , Ej , and then to take as the degree of the similarity of two elements, E(x,y), the minimum of all the degrees E j(x,y), which, as it is easy to check, is also a T-transitive relation. Obviously, there are other ways to combine fuzzy transitive relations which also preserve the transitive character of the relation. , any reflexive, symmetric and T –transitive fuzzy relation on a set X is generated by a family of fuzzy subsets of the given set through the procedure described in this section. In (Valverde and Ovchinnikov, 1986) it has been shown that the above representation also holds for left-continuous T –norms, this fact is specially interesting when the minimal T –norm Z is considered. As it is known, this T –norm is defined by
1,max,1
1,max,,min),(
yxif
yxifyxyxQ
Theorem. Let U be nonempty universal set, S a continuous t-conorm and m a mapping UU into <0,1>. Then m is pseudometric if, and only if there exist a
family njjh
1, such that
yhxhmyxm jjsj
j,sup),(
For some continuous and order reversing bijection on the unit interval.
In other words, any S-pseudometric on a given set U comes from a family of fuzzy subsets of the given set. So that, in the case of ordinary (bounded) metrics, the corresponding S- metric is yxyxm ),( . That is, once
a “distance” on the unit interval is fixed, this distance is carried to the given set U through the fuzzy subsets of U. Let it be noticed that such procedure is implicitely used in order to associate a likenes relation to a fuzzy partition. As it is known, at the very ®, a fuzzy partition of a set U was defined as a finite family of fuzzy subsets iu of U such that
1)( Ux
i xu , for any I and 0)( i
i xu , for any xU.
Definition: A function h from U to <0,1> is termed a generator of given T indistinguisability relation E , if Eh≥E,, HE will denote the set of all generators of E.
The next definition will play as important role in order to give a more convenient characterization of the generators of a T-indistinguishability relation E. It follows immediately from the representation theorem that, given a T-indistin-guishability relation E on U, the set UyyxE ),( of fuzzy subsets of X is
a generating family of E and will be denoted by Uyy xh
. The next definition
will play as important role in order to give a more convenient characterization of the generators of a T-indistinguishability relation E.
Definition. If E be T-indistinguishability relation then E is a map from <0,1>U into <0,1>U defined by yhyxETx
UyhE ,,sup
for any xU.
If U is a finite set then E is represented by a matrix and hE may be understood as the max-T product of E by the column vector representing the fuzzy set h.
Definition: A function h from U to <0,1> is termed a generator of given T indistinguisability relation E , if Eh≥E,, HE will denote the set of all generators of E.
The next definition will play as important role in order to give a more convenient characterization of the generators of a T-indistinguishability relation E. It follows immediately from the representation theorem that, given a T-indistin-guishability relation E on U, the set UyyxE ),( of fuzzy subsets of X is
a generating family of E and will be denoted by Uyy xh
. The next definition
will play as important role in order to give a more convenient characterization of the generators of a T-indistinguishability relation E.
Definition. If E be T-indistinguishability relation then E is a map from <0,1>U into <0,1>U defined by yhyxETx
UyhE ,,sup
for any xU.
If U is a finite set then E is represented by a matrix and hE may be understood as the max-T product of E by the column vector representing the fuzzy set.
Fuzzy Systems
Preference relations
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Preference relationsMore specifically, we define three important relations in A:
A couple of alternatives (a,b) belongs to the strict preference relation P if and only if the user prefers a to b;
A couple of alternatives (a,b) belongs to the indifference relation I if and only if the user is indifferent between alternatives a and b;
A couple of alternatives (a,b) belongs to the indifference relation I if and only if the user is indifferent between alternatives a and b;
A couple of alternatives (a,b) belongs to the incomparability relation J if and only if the user is unable to compare a and b, for instance caused by conflicting or insufficient information.
A preference structure on a set of alternatives A is the triplet (P,I,J) of a binary preference, indifference and incomparability relation in A. However, P, I and J must satisfy some rather basic additional conditions. For instance, any couple of alternatives belongs to exactly one of the relations P, Pt (the transpose of P), I or J. More formally, a preference structure is defined as follows.
Definition: A preference structure on a set of alternatives A is a triplet (P,I,J) of binary relations in structures since statements over degrees of and incompa-rability of preferences are natural and satisfy:
i. I is reflexive and J is irreflexive; ii. P is asymmetrical; iii. I and J are symmetrical; iv. P I =, P J = and I J = ; v. P Pt I J = A2
Example. Let A=a,b,c and
000
000
110
c
b
a
cba
P ,
010
100
000 ,
100
010
001
c
b
a
cba
J
c
b
a
cba
I
Then (P,I,J) is preference structure
Fuzzy preference relations
Definition. A triplet (P,I,J) of binary fuzzy relations in A is a fuzzy
preference relation on A if and only if
(i) I is reflexive or (P and J are irreflexive);
(ii) I is symmetrical or J is symmetrical
(iii) ( (a,b)A2)( P(a,b) + P(b,a) + I(a,b) + J(a,b) =1)).
Let A be a finite set of objects with at least two elements. We interpret the elements of A as alternatives among which a choice is to be made taking into account different points of view, e.g. several criteria or the opinion of several voters. A common practice in such a situation is to associate with each ordered pair (a, b) of alternatives a number indicating the strength or the credibility of the proposition “a is at least as good as b”, e.g. the sum of the weights of the criteria favoring a or the percentage of voters declaring that a is preferred or indifferent to b. This leads to a fuzzy (large) preference relation on A. In the area of ELECTRE III is a typical illustration of such a process. A fuzzy (binary) relation on a set A is a function R associating with each ordered pair of alternatives (a, b) A2 an element of 0, 1. Therefore, we define a choice procedure for fuzzy preference relations (on a set A) as function associating a nonempty subset of A, the “choice set”, with each fuzzy reflexive binary relation on A. In this note, we study “choice procedures” instead of the more classical notion of “choice functions”, i.e. functions associating a choice set with any subset of A. If a fuzzy relation R is such that R(a, b) {0, 1}, for all a, b A, we say that R is crisp. In this case, we write a R b instead of R(a, b) = 1.
Some properties of choice procedures
It is clear that an ordinal choice procedure does not make use of the cardinal properties of the numbers R(a, b). Many ordinal choice procedures can be envisaged. Let us mention one of them that has often been discussed in the literature and may be seen as a direct extension to the fuzzy case of the classical notion of the “greatest elements” of a crisp preference relation. Let
R F(A) and, for all a A, define, using the same notation as in Barrett et al. (1990), the ‘min in Favor’ score of alternative a letting:
),(min,
\caRRam
aAcF
A clearly ordinal choice I defined by RbmRamAaRC FFmF ,,; for all
bA.
. Let us illustrate the possibility of discontinuities on a simple example involving a crisp relation and an “almost crisp” one. Consider the relations
R a b c R´ a b c a 1 1 1 a 1 1 b 0 1 0 b 0 1 0 c 0 0 1 c 0 0 1
where 0 < λ < 1.
It is easy to see that R is crisp and that G® = {a}. Let C be a faithful choice
procedure. We have C® = {a}. Even if C is ordinal, it may happen that a ∉ C(R) whatever the value of λ. As a result C®∩C(R´) will be empty even when R is arbitrarily “close” to R. Our final axiom is designed to prevent such situations.
We say that this sequence converges to converges to R F(A2) if, for all ε , there is an integer k such that, for all j ≥ k and all a, b A, Rj(a,b)-R(a,b) . A choice procedure C is said to be continuous if, for all RF(A) and all sequences Rj F(A2) converging to R f(aC(Ri), for all Ri in sequence) aC® Our definition of continuity implies that an alternative that is always chosen with fuzzy relations arbitrarily close to a given relation should remain chosen with this relation. It is not difficult to see that C is continuous.
Fuzzy partial ordered relationsThe fuzzy relation is fuzzy partial ordered relation if it satisfy following
conditions
a) is reflexive(R(x,x)=1 for all xU)
b) is symmetric(If R(x,y)0 R(y,x)=0 for all xy)
c) is transitive(R(x,z)supminR(x,y),R(y,z) for all x,zU
Example: Fuzzy relation
1000
1100
9.07.010
8.06.05,01
R is fuzzy partial ordered relation
Note: Fuzzy relation R is fuzzy partial ordered relation if ad only if its -cut is
patial ordered relation for all 0,1.
Proof: We leave to reader.
Transitivity properties for fuzzy relations
We define the following transitivity conditions
1) Strong stochastic transitivity(S-transitivity)
minR(x,y ,R(y,z)0.5R(x,z)maxR(x,y),R(y,z)
2) Moderate stochastic transitivity
minR(x,y ,R(y,z)0.5R(x,z) minR(x,y ,R(y,z)
3) Weak stochastic transitivity
minR(x,y ,R(y,z)0.5R(x,z)0.5
4) -transitivity
minR(x,y ,R(y,z)0.5
R(x,z)maxR(x,y ,R(y,z)+(1-)minR(x,y ,R(y,z)
5) G-transitivity
R(x,z) R(x,y+R(y,z)-1
The G-transitivity is often called group transitivity because if n elements have
preference R which are linear ordered then
a) R(x,y)0,1
b) R(x,x)=0
c) R(x,y+R(y,z)=1 for xy
d) R(x,z) R(x,y+R(y,z)-1
It is well-known that from any reflexive binary relation R in a set of alternatives A, a classical preference structure (P,I,J) can be constructed in the following way:
(i) P = R ∩ coRt ;
(ii) I = R ∩ Rt ;
(iii) J = co R ∩ coRt.
Preference Structures Without IncomparabilityTheorem (Roubens and Vincke 1985). A preference structure (P,I,J) on U is a preference structure (P,I) on U if and only if its large preference relation is complete.
Two different types of fuzzy preference structures without incomparability can be distinguished.
Theorem (De Baets and Van de Walle 1995). A fuzzy preference structure (P,I,J) on U with fuzzy large preference relation R in U is a fuzzy preference structure (P,I) on A of Type 1 if and only if
(x,y)U2 maxR(x,y),R(y,x)=1
Theorem (De Baets and Van de Walle 1995). A fuzzy preference structure (P,I,J) on U with large fuzzy preference relation R in A is a fuzzy preference structure (P,I) on U of Type 2 if and only if
(x,y)U2 R(x,y)+R(y,x)1
In both classes, the following relationship between the fuzzy strict
preference relation P and the fuzzy large preference relation R holds: (x,y)U2 R(x,y)= 1-R(y,x)
Quasi-order relations and the analysis of preference relations
A binary (fuzzy) relation R in a universe U is called:
(i) reflexive if and only if xU2 R(x,x)= 1
(ii) a (fuzzy) quasi-order relation in U if and only if it is reflexive and transitive.
Theorem 4 (Fodor and Roubens 1994). Consider a binary fuzzy relation R in a universe U. R is a fuzzy quasi-order relation in U if and only if for all values of α (with α belonging to the interval <0,1>) it holds that Rα is a (crisp) quasi-order relation in U. The starting point of the analysis is the realization that every row in the matrix representation of a preference relation is a profile of the preferences a user has for an alternative compared to all other alternatives. The i-th row of P contains all preferences of the form P(xi,xj) the number of alternatives. Recall that we can denote the i-th row of P as the afterset aiP.
Fuzzy Systems
Fuzzy functions
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Fuzzy functionsOne of the fundamental conceptions of mathematics is the function f:AB
. It is nonempty binary relation fAB satisfying conditions
a) xA yB (x,y)f
b) (x1,y)f (x2,y)fx1=x2.
Let F(U) and F(V) are sets of all fuzzy sets on universes U,V. Then a fuzzy function U in V denoted by f:UV is a map
f: F(U) F(V)
If two fuzzy functions f a g are given
f:UV g:VW
the composition
g f:UW
Examlle: Let U={1,2} and V=R=(-,) and
5,0
5,4;5
4,3;3
3,0
f(2)
7,0
7,6;7
6,5;5
5,0
f(1)
x
xx
xx
x
x
xx
xx
x
Then f is fuzzy function from U={1,2} in V=R=(-,).
Definition: The fuzzy function from U in V, denoted by f:UV, is fuzzy subset of the product UV.
Example: A fuzzy function yxeyxf 2),( describes the statement x is approximatively equal 2y.
Example: Let A~
={R, xA
3,0
3,2,3
2,1,1
1,0
x
xx
xx
x
}, B~
={R, xB
4,0
4,3,3
3,2,1
2,0
x
xx
xx
x
}
Z=x+y. Then A =
0;
1,0(;3,1
R B =
0;
1,0(;4,2
R
If x 3,1 and y 4,2 then value membership function is more then
((z)
(y)}(x),min{max BA
zyxyx,
and that
4,2,3,1(y),(x) ,2,1,2,1BA yyxx
2743,2321 ,2,2,2,1,1,1 yxzyxz
C~
={R, zC
7,0
7,5,2
7
5,3,2
33,0
x
xx
xx
x
},
Note: Let A~
is fuzzy set of U and f is mapping U in V. Then usually projection A~
onto B~
Is fuzzy set with membership function
xy Axy
BA
max)(
Example: Let A~
={(-2,0.4), (-1,0.2),(0,1),(1,0,5), (2,0.8)} and y=x2. Then projection A~
is the fuzzy set B
~={(4,max{((-2)2,0.4), (22,0.8)}, (1,max{((-1)2,0.2),
(12,0.5)},(0,1)}={(0,1),}1,0.5),(4,0.8)}
Example: Let A~
={R, xA
3,0
3,2,3
2,1,1
1,0
x
xx
xx
x
} and y=x2
Then A =
0;
1,0(;3,1
R 2,1
222 13,1 yxy
,3,1 ,22,11 yy
If .9,10 21 yy If .4,41 21 yy and xy
9,0
9,4,3
4,1,1
1,0
y
yy
yy
y
FUZZY LOGIC
Fuzzy logic is used in system control and analysis design, because it shortens the time for engineering development and sometimes, in the case of highly complex systems, is the only way to solve the problem. Although most of the time we think of "control" as having to do with controlling a physical system, there is no such limitation in the concept as initially presented by Dr. Zadeh. Fuzzy logic can apply also to economics, psychology, marketing, weather forecasting, biology, politics ...... to any large complex system . Fuzzy logic is not the wave of the future. It is now! There are already hundreds of millions of dollars of successful, fuzzy logic based commercial products, everything from self-focusing cameras to washing machines that adjust themselves according to how dirty the clothes are, automobile engine controls, anti-lock braking systems, color film developing systems, subway control systems and computer programs trading successfully in the financial markets.
We are all familiar with binary valued logic and set theory. An element belongs to a set of all possible elements and given any specific subset, it can be said accurately, whether that element is or is not a member of it.
Unfortunately, not everything can be described using binary valued sets. The classifications of persons into males and females is easy, but it is problematic to classify them as being tall or not tall. The set of tall people is far more difficult to define, because there is no distinct cut-off point at which tall begins. This is not a measurement problem, and measuring the height of all elements more precisely is
not helpful. Such a proble
m is often
distorted so that it can be described using the
well known existing methodology. Here, one could simply select a height, e.g. 1.75m,
at which the set tall begins.
Fuzzy logic was suggested by Zadeh as a method for mimicking the ability of human reasoning using a small number of rules and still producing a smooth output via a process of interpolation. It forms rules that are based upon multi-valued logic and so introduced the concept of set membership. With fuzzy logic an element could partially belong to a set and this is represented by the set membership. For example, a person of height 1.79m would belong to both tall and not tall sets with a particular degree of membership. As the height of a person increases the membership grade within the tall set would increase whilst the membership grade within the not tall set would decrease. The output of a fuzzy reasoning system would produce similar results for similar inputs. Fuzzy logic is simply the extension of conventional logic to the case where partial set membership can exist, rule conditions can be satisfied partially and system outputs are calculated by interpolation and, therefore, have output smoothness over the equivalent binary-valued rule base. This property is particularly relevant to control systems.
Rules of propositional calculus
The following table lists some inference rules of propositional calculus The table makes use of mathematical notation. The following symbols occur in the table:
p q: p must be true, or q must be true (or both) p q: both p and q must be simultaneously true p q: p implies q: if p is true then so is q p q: p is logically equivalent to q: if either is true/false, then so is the other. p ├ q: from p infer q (by applying basic inference rules, q can be shown to hold
assuming p (note that this is equivalent to ( ⊢p → q). ┐p: not p
Basic arguments propositional calculus Name Sequent Description
Modus Pones [(p → q) p] ├ q if p then q; p; therefore q
Modus Tollens (p → q) ¬q] ⊢ ¬p if p then q; not q; therefore not p
Hypothetical syllogism
[(p → q) (q → r)] ├ (p → r)
if p then q; if q then r; therefore, if p then r
Disjunctive syllogism [(p q) ¬p] ├ q Either p or q; not p; therefore, q
Constructive dilemma
[(p → q) (r → s) (p r)] ├ (q s)
If p then q; and if r then s; but either p or r; therefore either q or s
Destructive dilemma
[(p → q) (r → s) (¬q ¬s)] ├ (¬p ¬r)
If p then q; and if r then s; but either not q or not s; therefore rather not p or not r
Simplification (p q) ├ p,q p and q are true; therefore p is true
Conjunction p, q ├ (p q) p and q are true separately; therefore they are true conjointly
Addition p ├ (p q) p is true; therefore, for any q, (p or q) is true
Composition [(p → q) (p → r)] ├ [p → (q r)]
If p then q; and if p then r; therefore if p is true then q and r are true
De Morgan´s theorem (1) ¬ (p q) ├(¬p ¬q)
If it is not true that p and q hold, then at least either p or q is not true
De Morgan's Theorem (2) ¬ (p q) ├ (¬p ¬q)
If it is not true that p or q holds, then p does not hold and q does not hold
Commutation(1) (p q) ├ (q p) (p or q) is equiv. to (q or p)
Commutation (2) (p q) ├ (q p) (p and q) is equiv. to (q and p)
Association(1) [p (q r)]├[(p q) r]
p or (q or r) is equiv. to (p or q) or r
Association (2) [p (q r)]├[(p q) r] p and (q and r) is equiv. to (p and q) and r (therefore, (p q ∧ r) is unambiguous)
Distribution (1) [p (q r)]├[(p q) (p r)]
p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2) [p (q r)] ├ [(p q) (p r)]
p or (q and r) is equiv. to (p or q) and (p or r)
Double negation p ├ ¬¬p p is equivalent to the negation of not p
Transposition (p → q) ├ (¬q → ¬p) If p then q is equiv. to if not q then not p
Material implication (p → q) (¬⊢ p ∨ q) If p then q is equiv. to either not p or q
Material equivalence (1)
(p↔q)├ [(p→q) (q→p)]
(p is equiv. to q) means, (if p is true then q is true) and (if q is true then p is true)
Material equivalence (2)
(p ↔ q)├ [(p q) (¬q ¬p)]
(p is equiv. to q) means, either (p and q are true) or ( both p and q are false)
Exportation
[(p q) → r] ├ [p → (q → r)]
from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
Importation [p → (q → r)] ├ [(p q) → r]
if r is true when q is true, under the condition that p is true, then if p and q are true, r is as well
Tautology p ├ (p ¬p) p is true is equiv. to p is true or p is false (this can be seen as a special case of addition)
The set of logic terms
Terms: The set of terms is recursively defined by the following rules:
1. Any constant is a term. 2. Any variable is a term. 3. Any expression f(t1,...,tn) of n ? 1 arguments (where each argument ti is a term
and f is a function symbol of valence n) is a term. 4. Closure clause: Nothing else is a term.
Well-formed formulas
The set of well-formed formulas (usually called wffs or just formulas) is recursively defined by the following rules:
1. Simple and complex predicates If P is a relation of valence n ? 1 and the ai are terms then P(a1,an) is well-formed. If equality is considered part of logic, then (a1 = a2) is well formed. All such formulas are said to be atomic.
2. Inductive Clause I: If φ is a wff, then ¬φ is a wff. 3. Inductive Clause II: If φ and ψ are wffs, then (φ ψ) is a wff. 4. Inductive Clause III: If φ is a wff and x is a variable, then ¬x φ is a
wff.
5. Closure Clause: Nothing else is a wff.
Free VariablesFree Variables:
1. Atomic formulas if φ is an Atomic formula then x are free in φ if and only if x occurs in φ.
2. Inductive Clause I: x is free in ¬φ if and only if x is free in φ. 3. Inductive Clause II: x is free in (φ ? ψ) if and only if x is free in φ or x is free in
ψ. 4. Inductive Clause III: x is free in ? y φ if and only if x is free in φ and x y. 5. Closure Clause: if x is not free in φ then it is bound...
Since ¬ (φ ¬ψ) is logically equivalent to (φ ψ), (φ ψ) is often used as a short hand. The same principle is behind (φ ψ) and (φ ψ). Also x φ is a short hand for ¬y ¬φ. In practice, if P is a relation of valence 2, we often write "a P b" instead of "P a b"; for example, we write 1 < 2 instead of <(1 2). Similarly if f is a function of valence 2, we sometimes write "a f b" instead of "f(a b)"; for example, we write 1 + 2 instead of +(1 2). It is also common to omit some parentheses if this does not lead to ambiguity.
Sometimes it is useful to say that "P(x) holds for exactly one x", which can be expressed as! x P(x). This can also be expressed as !x (P(x) y (P(y) ? (x = y))).
Substitution
If t is a term and φ(x) is a formula possibly containing x as a free variable, then v φ(t) is defined to be the result of replacing all free instances of x by t, provided that no free variable of t becomes bound in this process. The problem is that the free variable y of t (=y) became bound when we substituted y for x in φ(x). So to form φ(y) we must first change the bound variable y of φ to something else, say z, so that φ(y) is then z z ? y.
EqualityThe most common convention for equality is to include the
equality symbol as a primitive logical symbol, and add the axioms for equality to the axioms for first order logic. The equality axioms are
1. x = x 2. x = y f(...,x,...) = f(...,y,...) for any function f 3. x = y (P(...,x,...) ? P(...,y,...)) for any relation P
(including = itself)
A sentence is defined to be provable in first order logic if it can be obtained by starting with the axioms of the predicate calculus and repeatedly applying the inference rules "modus ponens"
Axioms of basic logic
Let ,, are formulae then A1: () ( ) ( ) A2: ( ) A3: ( ) ( ) A4: () ( ) A5a: ( ) ( ) A5b: ( ) ( ) A6: ( ) (() ) A7: In classic logic the axiom are Let ,, are formulae then
1) ( ) 2) ( ) () ( ) 3) ( ) ( )
The membership functions predicates into fuzzy logic
K l e e n e - D i e n e s i m p l i c a t i o n
y,x)y(,x1max)y,x( 1BA1R
L u k a s i e w i c z i m p l i c a t i o n
y,x)y(x1,1min)y,x( 2BA2R
Z a d e h i m p l i c a t i o n
y,xx1,)y(,xminmax)y,x( 3ABARm S t o c h a s t i c i m p l i c a t i o n
y,xx)y(x1,1min)y,x( 4ABARs
G o g u e n i m p l i c a t i o n y,x
)y(
x,1min)y,x( 5
B
ARA
G o e d e l i m p l i c a t i o n y,x
inak),y(
)y(x,1n)y,x( 6
B
BARg
S h a r p i m p l i c a t i o n y,x
inak,0
)y(x,1)y,x( 7
BARI
M a m d a n i i m p l i c a t i o n y,x)y(,xmin)y,x( 8BARM
L a r s e n i m p l i c a t i o n y,x)y(.x)y,x( 9BARL
Fuzzy Systems
Fuzzy numbers
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Fuzzy numbersA fuzzy set is fuzzy convex set -cut is convex for all 1,0 .
A fuzzy set is normal if 1, xx .
Let U is set of real number. A fuzzy number is a convex normal fuzzy set ,~RA
whose membership function is at least segmentally continuums. If babaxx ,,1, they also fuzzy number call fuzzy interval.
Triangular fuzzy number
A fuzzy number is triangular fuzzy number if its membership function is
cx
cbxbc
xc
baxab
axax
x
,0
,,
,,
,0
or if its horizontal representation is
0,0
1,0(,,, 21
bccabaxxA
A triangular fuzzy number is often called fuzzy number.
Triangular fuzzy number
Trapezoidal fuzzy numberA fuzzy number is trapezoidal ( or ) if its membership function is
dx
dcxcd
xdcbx
baxab
axax
x
,0
,,
,,1
,,
,0
or if its horizontal representation is
0,0
1,0(,,, 21
cddabaxxA
Trapezoidal fuzzy number
S and Z fuzzy numbers
A trapezoidal fuzzy number is often expressed as (a,b,c,d). The triangular number as (a,b,c). If b=c a trapezoidal fuzzy number is triangular.
Some typical fuzzy numbers
A fuzzy number is positive if cut Afor all 0. A fuzzy number is negative if cut A for all 0. If (0)0 the fuzzy number is fuzzy zero.
Representation of fuzzy numberT h e o re m : L e t fo r a ll 1,0( c u t o f f u z z y n u m b e r A
~ i s c lo s e d in te r va l . T h e n t h e re
e x i s t n u m b e rs a b cd a n d f u n c tio n s u , v t h a t
,,
,,1
,,
cxxv
cbx
bxxu
x
a n d u ( x )= 0 fo r a l l x ( - ,a ) a n d v( x) = 0 fo r a ll x (d , ) . P ro o f: It i s e v id e n t. S e e p ic t u re .
Supremum and infimum of fuzzy number
L e t BA~
,~
a r e f u z z y n u m b e r s . T h e n f u z z y n u m b e r C~
i s t h e i r
i n f i m u m C~
= BA~~ i f t h e m e m b e r s h i p f u n c t i o n o f C
~ i s
Ryxyxzyxx BA ,;,min,minsup
T h e f u z z y n u m b e r C
~ i s t h e i r s u p r e m u m C
~ = BA
~~ i f t h e m e m b e r s h i p f u n c t i o n o f C
~ i s
Ryxyxzyxx BA ,;,max,minsup
Comparable fuzzy numbers
The fuzzy number BA~~
(A~ is greaterB
~) if B~= BA
~~ or A
~= BA
~~ .
Note BA~~
if xx BA for all xU.
The fuzzy numbers BA~
,~
are comparable if BA~~
or BA~~
.
If BA~~
or BA~~
is false then BA~
,~
are not comparable.
Zadeh's extension principle
You can use fuzzy numbers for fuzzy arithmetic. This can be done by the application of Zadeh's extension principle.
In the cartesian product of two fuzzy numbers A and B you take the MINIMUM of the grades of membership of the two corresponding sub-numbers ai and bi that are operated on, to determine the grade of membership of the new sub-number ci resulting from that operation.
Then you take the MAXIMUM of the grades of membership of the subnumbers with the same numerical value ci to determine the grade of membership of the sub-number ci of the new fuzzy number C. In short it's the "MAX of MIN's".
Operation addition and difference
T h e o p e r a t i o n s o n t r i a n g u l a r f u z z y n u m b e r s a r e f r e q u e n t l y d e f i n e d a s a n o p e r a t i o n s w h i c h r e s u l t i s t r i a n g u l a r f u z z y n u m b e r .
babbbaaacccc~~,,,,,,~
321321321
I f o p e r a t i o n i s a d d i t i o n t h e m
bababababbbaaacccc~~,,,,,,,,~
332211321321321
I f o p e r a t i o n i s d i f f e r e n c e t h e m
bababababbbaaacccc~~,,,,,,,,~
132231321321321
Interval arithmeticLet dcba ,,, are two intervals. Then arithmetical operations arte defined:
Addition:
dbcadcba ,,,
Difference:
dbcadcba ,,,
Multiply:
bdbcadacbdbcadacdcba ,,,max,,,,min,,
Division:
cdbadcba
1,
1,,/,
If c,d are positive numbers or negative.
Operation multiplication and division
If operation is multiplication them
bababababbbaaacccc~~,,,,,,,,~
332211321321321
If and only if all fuzzy numbers are positive
If operation is division them
31
3
2
2
3
1321321321
~,,,,/,,,,~
b
a
b
a
b
a
b
abbbaaacccc
If and only if all fuzzy numbers are positive.
Example
Let 5,3,2~
,4,3,1~ ba then 4,21a , 25,2 b and
a, b =a1,b1+a2,b2= a1+a2, b1+b2= 25,24,21 =
= 39,33
a, b =a1,b1-a2,b2= a1-b2, b1-a2= 25,14,21 =
= 23,44 .
a, b=a1,b1a2,b2=a1,b1.a2,b2=
25,1.4,21
= 22 21320,231
Extension principleLet A
~, B~
, Z~
are fuzzy numbers and A, B, Z their membership functions. Then
membership function for Z~
= A~
+ B~
is defined as
yxzyBxAzZyx
)(),(minsup)(,
The membership function for Z~
= A~
- B~
is defined as
yxzyBxAzZyx
)(),(minsup)(,
The membership function for Z~
= A~
. B~
is defined as
yxzyBxAzZyx
)(),(minsup)(,
The membership function for Z~
= A~
/ B~
( B~
is non zero number)is defined as
yxzyBxAzZyx
/)(),(minsup)(,
Fuzzy Systems
-cuts and interval arithmetic
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Interval arithmetic Let dcba ,,, are two intervals. Then arithmetical operations arte defined:
Addition:
dbcadcba ,,,
Difference:
dbcadcba ,,,
Multiply:
bdbcadacbdbcadacdcba ,,,max,,,,min,,
Division:
cdbadcba
1,
1,,/,
If c,d are positive numbers or negative.
Example
Let I1=2,3, I2=-5,-3. What is sum, difference, product and fraction of I1,I2? I1+I2= 2,3+-5,-3=-3,0. I1-I2= 2,3--5,-3=2,3+3,5=5,8. I2-I1= -5,-3-2,3=-5,-3+-3,-2=-8,-5. I2-I2= -5,-3--5,-3=-5,-3+3,5=-2,2. I1.I2= 2,3.-5,-3=-15,-9.
5
2,1
5
2,
3
3
5
1,
3
13,23,5/3,2I/I 21
1,
2
5
3
3,
2
5
2
1,
3
13,53,2/3,5I/I 12 .
-cuts and interval arithmetic
Let A~
is fuzzy number ad all -cuts of A~
are closed interval. Then its horizontal representation is set of closed intervals a1 (), a2 () , (0,1 and we can define operations on fuzzy numbers as operation of interval arithmetic.
Addition of fuzzy numbers
Then membership function for Z~
= A~
+ B~
is defined as
yxzxBxAzZyx
)(),(minsup)(,
Horizontal representation of this operation is
212121 ,,, bbaazz
Difference of fuzzy numbers
Then membership function for Z~
= A~
- B~
is defined as
yxzxBxAzZyx
)(),(minsup)(,
Horizontal representation of this operation is
212121 ,,, bbaazz
Product of fuzzy numbers
Then membership function for Z~
= A~
. B~
is defined as
yxzxBxAzZyx
.)(),(minsup)(,
Horizontal representation of this operation is
212121 ,.,, bbaazz
Fraction of fuzzy numbers
Then membership function for Z~
= A~
/ B~
is defined as
yxzxBxAzZyx
/)(),(minsup)(,
Horizontal representation of this operation is
212121 ,/,, bbaazz
Function on fuzzy numbers
A fuzzy function is a mapping from fuzzy numbers into fuzzy numbers. We write h( x~ )= y~
for a fuzzy function with one independent variable x~ . For two independent variables we have h( x~ , y~ )= z~ . Let h:a,bR. We extend h( x~ )= y~ in two ways
a) the extension principle b) -cuts and interval arithmetic.
Extension principle
Let h:a,bR and x~ is fuzzy number(usually triangular or trapezoidal). Then membership function is
bazzxhxxzyzxhx
,,)(;~sup~)~(
ExampleLet h(x)=x2 and x~ is triangular fuzzy number (-1,1,2). What is h( x~ )? It is clear h( x~ ) is fuzzy number. Membership function of x~ is
2,1;2
1,1;12
12,1,0
)(~
xx
xx
x
xx
The membership function of h( x~ )= y~ is
2,1,;~sup~)~( 2 xzxxxzyzxhx
The support of x~ is -1,2 and the support of y~ is 0,4,
x~ (x)-1,0 x~ (x)0,1 and
y~ (z)= 12
1z ,z0,1
and y~ (z)= z2 ,z1,4
-cuts and interval arithmetic
If h is continuous, then we can find -cuts of y~ . Let y~ ()=y1(),y2() .
Where
xxxhy ~)(min1
xxxhy ~)(max2 ,0,1
Example
Let h(x)=x2 and x~ is triangular fuzzy number (-1,1,2). What is h( x~ )? It is clear h( x~ ) is fuzzy number. Membership function of x~ is
2,1;2
1,1;12
12,1,0
)(~
xx
xx
x
xx
-cuts of x~ are -1+2,2-
15.0,21
5.0,0
2,21min~)(min
2
21
xxxxxhy
222 22,21max~)(max xxxxxhy ,
0,1
Metrics on fuzzy numbersIf x,y are real numbers, then their distance is d=x-y an is contents a of parallelogram(see fig). We use this geometrical notation to define a distance of two fuzzy numbers.
L e t f :A , B C , D is a m a p inte rva l A , B o nto i nte rva l C , D . T he n
BAyxfA
yxfxyf BAx
, xallfor );
;sup,1
is pse ud o in ve rse fu nc tio n to f.
Metrics on fuzzy numbers
Let ba~,~ are two fuzzy numbers and xLxL ba , are left parts
their membership functions and xPxP ba , are right parts their membership functions. Then
dyyPyPyLyLbad baba )(, 111
0
11
is distance of fuzzy numbers ba~,~ .
Metrics on fuzzy numbers
Infimum and supremum of fuzzy numbers
Let ba~
,~ are two fuzzy numbers then
Ryxyxzyxzz bababa ,,,min,minsup,inf
Ryxyxzyxzz bababa ,,,max,minsup,sup
Infimum ba~~
Supremum ba~~
Comparable fuzzy numbers
Let ba~
,~ are two fuzzy numbers then fuzzy number a~ is greater or equal b~
if
a~ ba~~ or b
~ba~~
Fuzzy Systems
Fuzzy linear equation
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Fuzzy linear equation
L e t xcba ~,~,~
,~ a re fu z z y n u m b e rs th e n
cbxa ~~~~
is fu z z y l in e a r e q u a t io n . P ro b le m : H o w to s o lve it?
Major problem in solving fuzzy equations
If a, b, c, xR and a0 then a
bcx
, but 0
~~ bb and 1~/~ aa
Example: Let a~ is triangular fuzzy number (a, b, c) then
a~ -a~=(a-c,0,c-a)(0,0,0) and 1,1,~/~
a
c
c
aaa . For instance if a~ =(1,2,3)
then a~ -a~=(-2,0,2) and 13,1,3
1~/~
aa .
This is a major problem in solving fuzzy equations.
Classical method solving of fuzzy equation
Let
xcba ~,~,~
,~
are -cuts of xcba ~,~,~
,~ . Then fuzzy equation we can express as
cbxa ~~~~
If
2121
2121
,~,,~,,
~,,~
xxxccc
bbbaaa
21212121 ,,,, ccbbxxaa
This solution (when it exists) we denote as cx~
Necessary conditions
If classical solution cx~ 21 ,xx exists then
1. 1x is monotonically increasing 2. 2x is monotonically decreasing 3. 1x 2x for all 01
Example cx~ does not exists
Let 3,2,1~a , 1,2,3~ b , 5,4,3~c Then its -cuts are
3,1, 21 aa , 1,3,21 bb
5,3,21 cc
and 21212121 ,,,, ccbbxxaa is
21222111 ,, ccbxabxa
5,313,31 21 xx
3
6,
1
621 xx
1x is decreasing and so cx~ does not exists.
Example existscx~
Let 3,2,1~a , 1,2,3~ b , 3,0,3~ c Then its -cuts are
3,1, 21 aa , 1,3, 21 bb
33,33, 21 cc
and 21212121 ,,,, ccbbxxaa is
21222111 ,, ccbxabxa
33,3313,31 21 xx
3
22
3
24,
1
22
1
221 xx
1x is increasing, 2x is decreasing and
2642623
22
1
22
and so cx~ exists.
Extended principle of solution fuzzy equation
L e t xcba ~,~,~
,~ a r e f u z z y n u m b e r s a n d cbxa ~~~~ i s f u z z y e q u a t i o n t h e n i t t o o
o f t e n h a s n o t s o l u t i o n . T h e f u z z i f i e d c r i s p s o l u t i o n i s abcx ~/)~~(~ . W e c a n
e v a l u a t e t h i s f o r m u l a i n t w o w a y s . 1 . e x t e n s i o n p r i n c i p l e 2 . - c u t s a n d i n t e r v a l a r i t h m e t i c .
I f ex~ i s e v a l u a t e d b y e x t e n s i o n p r i n c i p l e t h e n i t s m e m b e r s h i p f u n c t i o n i s
xabccbauxx e /,,max~
w h e r e
ccbbaacbau ~,~
,~min),,(
-cuts and interval arithmetic of solution fuzzy
I f t h e r e s u lt i s Ix~ th e n
a
bcx I ~
~~~
T h e o r e m : If e x i s ts t h e n T h e o r e m : N o t e : F o r m o r e c o m p li c a te d f u z z y e q u a t i o n s w i ll b e d i f f i c u l t to c o m p u te . F o r t h i s r e a s o n w e s u g g e s t a p p r o x i m a t i n g b y a n d ,
cx~ cx~
cx~
ex~
ex~
ex~
Ix~
Ix~
An example of solution fuzzy equation
L e t 3,2,1~ a , 1,2,3~ b , 5,4,3~ c T h e n i ts - c u ts a re
3,1, 21 aa , 1,3, 21 bb
5,3, 21 cc
1
28,
3
24
3,1
1,35,3~
~~~
a
bcx I
Evaluating of fuzzy formulasLet f: ARR is real function of real variable(s). We usually compute its value apply finite number of basic arithmetic operations. Fur instance
50401206)sin(
753 xxxxx
The image of fuzzy number x~ is evaluated in two methods 1. extension principle 2. -cuts and interval arithmetic. If it is used extension principle then membersip function of fuzzy number
xfy ~~ is
zxfxxzyx
~sup~
If f is continuous, then -cuts of xfy ~~ is
xxxfxxxfyyy max,min, 21
Alpha-cuts and interval arithmetic
All the functions we use in engineering have algorithm which use finite number of basic arithmetical operations. For instance
50401206)sin(
753 xxxxx
In fuzzy mathematic we have the interval x and we perform needed operations
interval arithmetic. For instance
50401206
)sin(753 xxx
xx
Example
Let f(x)=x(1-x)=y, xxxfy ~1~~~ and 21 , xxx . Let x~ is triangular fuzzy number (0,0.25,0.5).
Extension principle xfy ~~ and is
42
1,
4
x
xxxfxxxfyyy max,min, 21
xxxxxxxxy )1(max,)1(min =
=16
4,
16
4 22
-cuts and interval arithmetic
Let f(x)=x(1-x)=y, xxxfy ~1~~~ and 21 , xxx . Let
x~ is triangular fuzzy number (0,0.25,0.5). Then -cut of x~ is 42
1,
4
x and
42
1,
41
42
1,
41
xxy
*22
16
68,
16
2
41,
42
1
42
1,
4y
Fuzzy Systems
Fuzzification and defuzzification
Slovak University of TechnologyFaculty of Material Science and Technology in Trnava
Fuzzification and defuzzification
As a result of applying the previous steps, one obtains a fuzzy set from the reasoning process that describes, for each possible value, how reasonable it is to use this particular value. In other words, for every possible value, one gets a grade of membership that describes to what extent this value is reasonable to use. Using a fuzzy system as a controller, one wants to transform this fuzzy information into a single value that will actually be applied. This transformation from a fuzzy set to a crisp number is called a defuzzification. It is not a unique operation as different approaches are possible. The most important ones for control are described in the following.
Center of area or center of gravity method (COG)
This approach has its origin in the idea to select a value that, on average, would lead to the smallest error in the sense of a criterion. If is chosen, and the best value is x then the error is x-. Thus, to determine x the least squares method can be used. As weights for each square (x-)2, one can take the grade of membership with which x is a reasonable value. As a result one has to find
U U
au
aadxxxadxxxdxaxx 222
)(2)(min)(min
U
adxxx 02)(2
U
dxxxa
The center of area or center of gravidity is
U
U
dxx
dxxx
Center of area or center of gravity method.Discrete fuzzy set
If fuzzy set is discrete then
U UaUa
axxaxxaxx 222 )(2)(min)(min
U
U
x
xx
)(
Example
Let
4,0
4,2;2
42,1,1
1;0
x
xxx
x
x . Then defuzzificated value is
9
29
2
36
29
12
1
46
5
2
41
2
41
2
1
4
2
2
1
4
2
xdxx
dxx
xdxx
xdxx
dxx
dxxx
U
U
Center of sum (COS)
The defuzzification can be strongly simplified if the membership functions of the conclusions are singly defuzzified for each rule such that each function is reduced to a singleton that has the position i of the individual membership function's centre of gravity. The centre of singletons is calculated by using the degree of relevance as follows:
ii
iii
s
s
The simplification consists in that the singletons can be determined already during
the design of the fuzzy system and that the membership function with its complicated geometry is no longer needed. The defuzzification using this formula is an approximation of the defuzzification. Experiences from control show that there are slight differences between both approaches, which can be in most cases neglected.
First of maximum methods (FoM)
This class of methods determines by selecting the membership function with the maximum value. If the maximum is a range, the lower, upper or the middle value is taken for depending on the method. Using these methods, the rule with the maximum activity always determines the value, and therefore they show discontinuous and step output on continuous input. This is the reason why these types of method are not attractive for use in controllers.
Last of maximum methods (LoM)This class of methods determines by selecting the
membership function with the last maximum value. If the maximum is a range, the upper or the middle value is taken for depending on the method. Using these methods, the rule with the maximum activity always determines the value, and therefore they show discontinuous and step output on continuous input. This is the reason why these types of method are not attractive for use in controllers.
Margin properties of the centroid methods
As the centre of gravity of the area below the membership functions cannot reach the margins of x, the membership functions, which are at the margins, must be symmetrically expanded when obtaining the centre of gravity. This is necessary in order to have the full range of x available.
Margin of (a) original and (b) expanded
The methods of defuzzification
RCOM (random choice of maximum) FOM (first of maximum) LOM (last of maximum) MOM (middle of maximum) COG (center of gravity) MeOM (mean of maxima) BADD (basic defuzzification distributions) GLSD (generalized level set defuzzification) ICOG (indexed center of gravity) SLIDE (semi-linear defuzzification) FM (fuzzy mean) WFM (weighted fuzzy mean) QM (quality method)
The methods of defuzzification EQM (extended quality method) COA (center of area) ECOA (extended center of area) CDD (constraint decision defuzzification) FCD (fuzzy clustering defuzzification)
The maxima methods are good candidates for fuzzy reasoning systems. The distribution methods and the area methods exhibit the property of continuity that makes them suitable for fuzzy controllers .
Defuzzification: criteria and classification, from the journal Fuzzy Sets and Systems, Van Leekwijck and Kerre, Vol. 108 (1999), pp. 159-178
Linguistic Variable
Linguistic Variable - Linguistic means relating to language, in our case plain language words. Speed is a fuzzy variable. Accelerator setting is a fuzzy variable. Examples of linguistic variables are: somewhat fast speed, very high speed, real slow speed, excessively high accelerator setting, accelerator setting about right, etc.
A fuzzy variable becomes a linguistic variable when we modify it with descriptive words, such as somewhat fast, very high, real slow, etc. The main function of linguistic variables is to provide a means of working with the complex systems mentioned above as being too complex to handle by conventional mathematics and engineering formulas.
Linguistic variables appear in control systems with feedback loop control and can be related to each other with conditional, "if-then" statements. Example: If the speed is too fast, then back off on the high accelerator setting.
Universe of Discourse
Universe of Discourse - Let us make women the object of our consideration. All the women everywhere would be the universe of women. If we choose to discourse about (talk about) women, then all the women everywhere would be our Universe of Discourse.
Universe of Discourse then, is a way to say all the objects in the universe of a particular kind, usually designated by one word, that we happen to be talking about or working with in a fuzzy logic solution.
A Universe of Discourse is made up of fuzzy sets. Example: The Universe of Discourse of women is made up of professional women, tall women, Asian women, short women, beautiful women, and on and on.
Fuzzy Algorithm Fuzzy Algorithm - An algorithm is a procedure, such as
the steps in a computer program. A fuzzy algorithm, then, is a procedure, usually a computer program, made up of statements relating linguistic variables.
Examples:
If "green x" is very large, then make "tall y" much smaller.
If the rate of change of temperature of the steam engine boiler is much too high then turn the heater down a lot.
Zadeh's original definition of a linguistic variable is rather inspired by computational linguistics and classical AI and much more sophisticated than the shallow understanding that is most often used in engineering-oriented domains like fuzzy control.
Linguistic VariableUsually, a linguistic variable is a quintuple (L, G, T, U, S), where L, T, U, G, and S are defined as follows: 1. L is the name of the linguistic variable V (label) 2. G is a grammar 3. T is the so-called term set, i.e. the set linguistic expressions resulting from G 4. U is the universe of discourse 5. S is a T F(X) mapping which defines the semantics - a fuzzy set on X -of each linguistic expression in T.
