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Знания-Онтологии-Теории (ЗОНТ-09)
About a theory of canonically conjugate fuzzy
subsets
T.Gachechiladze 1, H.Meladze2, G.Tsertsvadze3, N.Archvadze4, T.Davitashvili5
1,4,5.
Department of Computer Sciences, Iv. Javakhishvili Tbilisi State University , 2, University str.,
0143, Tbilisi, Georgia
2.
Department of Computer Sciences , St Andrea First Georgian University, I.Chavchavadze str. 53-a,
3
N.Muskhelishvili Institute of Computational Mathematics, Akuri str №8 a, Tbilisi, Georgia
[email protected], [email protected],
[email protected], [email protected], [email protected]
Resume. The theory of canonically conjugate fuzzy subsets is presented. The Heizenberg’s
principle’s analog principle is established. In Hilbert space the informational functions and joint
membership functions are defined. In the work colour operators, Zadeh operators and
corresponding commutativity relations are presented.
Keywords. Fuzzy sets, membership function, colour operator, Zadeh operator.
1 Introduction
In many cases there exist unlimited number of ways of interaction of subject with the object. As a
result the controlled interaction is almost always incomplect. It is based on the limited (generally
small) number of attributes (colour) of the object corresponding to the subject, which can be
recognized. When the set of colours are defined as a result of our interaction with the object we say,
that there is defined the system on the object with the given structure of uncertainty [1].
Data in informatics are the set of so-called informational units [2]. Each informational units is the
four: (object, sing, value, verity). It is important to differ the notion of inaccuracy and uncertainty.
Inaccuracy is connected with the information content (corresponding component - “value”) and
uncertainty to its verity, understandable in terms of compatibility with reality (component “verity”).
For the given information there exist contradiction between inaccuracy of expression content and its
uncertainty expressed in the fact that with the increase of expressions accuracy its uncertainty rise and
vise versa, uncertain character of information leads to some inaccuracy of the final conclusions
received from this information. We see that from one side these notions are in certain contradiction and
from another side – complete each other.
We offer to model such situation by means of a new concept of optimal pare of canonicaly
conjugate fuzzy subsets. We offer the method of construction of the informational unit membership
functions taking into account the both canonicaly conjugate components simultaneously and have
discribing this unit in the most complete and optimal way.
2 Problem statement
Let  denotes some colour,  - his numerical value is random variable. In the referential system
 (universal set) it is latent parameter.
Let probability distribution of  is characterized by density (x):
  ( x )dx  1 .
R
Quantity
x*  M   x  ( x )dx
(1)
R
will be called calculated value of the colour  in point  of universal set . Note that formula (1)
establish one to one corresponds under  and real numbers R.
Except the quantity M , the colour of  is characterized by dispersion
 2 ( )   x  x*    ( x )dx .
2
R
In our model
  () is connected with definition of presence of colour  for . If  2    0 ,
2
 2 () , the more  is uncertain in . If
we will say, that  has sertain value in . The more is
 2     , we say that  has no colour . Thus if     1 we say that  possess colour , if
    0 , than  does not possess colour .
3 Problem solution
For    introduce some interval of  values
~   

I  ( )
I    R by relation
 ( )dx   I  ( )  ( x )dx
R
where I    is defined in a such way, that (4) will be true when
   is a membership function
established by expert. In the theory of information representation main role plays the notion of
informational function which is defined by
x ;    ( ) e i ,
where
 is the arithmetic root and  - random phase (real number). It is evident, that
 ( )  x ; 
Here x; 

 x; 
and

x;  .
x;  are bra- and ket- vectors [3] correspondingly.
Let consider the Fourier transform of the ket -vector:

 F x ;  
xc ; c
1
2c

x ; 
i
 x xc
e c
dx

(2)
R
where c is a constant. Integral (2) converges in mean-quadratic sense to the function from L2(R)

(Hilbert
space).
Correspondence
under
and
is
reciprocal:
F x ; 
x ; 

F : F 1 x ;   x ;  . If in L2(R) the scalar product is defined, than
 x ; 1 , x ; 2    xc ; c1 , xc ; c2  .
1
2
1
2
(3)
From formula (3) we can conclude that in parallel with colour  exist other colour c with
informational function xc ; c and density  (c ) which determines corresponding membership
c
function:
2

 c (c ) 
x c ; c x c ; c d c  I c (c ) x c ; c dc
I  c (c )
R


where I (c ) is the interval corresponding to membership function of canonically conjugate fuzzy
c
subset.
4 Simultaneouse distribution of canonicaly conjugate fuzzy subsets
Usually fuzzy subset is constructed on the basis of expert estimation of one of the noncommuting
component. Fuzzy subset constructed in this way characterize the information unit incompletely. We
offer the method of constructions of the informational unit membership function taking into account
the both canonically conjugate components simultaneously and hence describing this unit in most
complete and optimal way.
The joint distribution of two canonically conjugate colours is given by following formula:
  c ( x , xc ) 
1
4 2
  M (1,  2 ) exp( i(1x   2 xc )d1d 2
(4)
RR
and corresponding joint membership function is:
 ( x , xc ;  , c 

  c ( x , xc )dx dxc ,
I  ( ) I c (c )
(5)

where (4) is the Weil transformation [4] of the operator M 1 , 2  :

M 1 ,  2  
Operator

dx dxc   c x , xc x , xc  .
2 c  
1
3

x , xc  has the form:

i
1


 x , xc    dv exp 
xc v  x  v
2
 2c

1
x  v .
2
5 Optimal canonicaly conjugate fuzzy subsets
From formulas (4) and (5) follows very important relation between dispersions of canonically
conjugate colours:



c2
  ( )    D ( )  c (c )    D (c )  ,
c
4
2
2
2
2
(6)
where D and cD denotes dual colours.
This relation is analogue of Heisenberg’s uncertainty principle [5]. Generalized information theory
leaves to interpret the constant c in terms of the Shanonn measure of uncertainty.
When in formula (6) one have equality this means that corresponding membership function is
optimal. In presented work the form of optimal informational and membership functions are
established.
 
Formula, which corresponds to (4) with A  M 1, 2  have following form:
W  c  x , xc  
1
2

x  cv exp ivxc  x  cv dv
or
W  c x , xc  

1
2
exp  i c

x xc
2 c


 x ; xc ;c exp 1 xc xc 

 2c


Canonicaly conjugate fuzzy subsets with minimal value of product
are called optimal fuzzy subsets.
It can be demonstrated, that Wigner function in this case is
W~opt
~c
x , xc    c 
1


 x  x*
exp   2
2 


Notice, that W~opt~ is the product of optimal functions
  c

2
~

  ~2 D   ~2   ~2 D 
c 
 c
2  2xc  xc* 2  2 
c2





 x  x* 2 


x  
exp   2  and
2


2

2 




* 2
1
 xc  xc  ,
opt
~ c xc  
exp 


2 2
22
c


c
1
opt
~


where
 2   ~2   ~2 D
and

. x* 
 2   ~2   ~2 D 
c
c
c
c2
4 2
1
 ( x  x * )2 
2 2 ( w)  | xw  xw* | exp   w 2 w 
 2  ( w) 
R
Corresponding optimal compatibility function have the form:
opt
~~  , c  
c

 W~~c x , xc dx dxc
opt
I~  I~c
The characteriistic intervals
I  x*  1 ( )  ; x*

  2 ( ) 

,

I c  xc*  1c (c )  c ; xc*   2c (c )  c .
In these formulas
1 (),  2 ( ), 1c (c ), 2c (c )  R
and are determined by the
structure of supports of canonically conjugate fuzzy subsets.
6 Illustrative examples
As illustration what we said previously, cousider arithmetic operations on canonically conjugate
optimal fazzi real numbers. In the basis of optimal arithmetics laid correspondence between main
error and standart deviation:
x | x  x* | .
Mathematical expectation of this quantity call error of counted value
x* 
 ( x  x* ) 2 
2
*
|
x

x
|
exp
 ( )

 dx 


2

2
 
2 ( ) R
 2  ( ) 
1
Errors of arithmetic operations as it is known are defined by
( x1*  x2* )  ( x1* )  ( x2* ) ,
( x1*  x2* ) | x1*x2*  x2*x1* | ,
 x *  x *x *  x *x *
  1*   2 1 *2 1 2 ,
x2
 x2 
Because of (7) for arithmetic operations on optimal fuzzy numbers we have following relations:
 ( x1*  x2* )  ( x1* )  ( x2* )
 ( x1*  x2* ) | x1* ( x2* )  x2* ( x1* ) |
 (x  x ) 
*
1
*
2
x2  ( x1* )  x1*  ( x2* )
x2*
(7)
Connection between canonically conjugate colours will be find in the case of arithmetic
operations. For
 
C
c
2 
one can write in
C -gange
following formulas:
  ( xC* 1 )  ( xC* 2 )
 (x  x ) 
  ( xC* 1 )    ( xC* 2 )
*
C1
c
*
C1
C
C
C
  ( xC* 1  xC* 2 ) 
C
C
  ( x )  ( xC* 2 )
xC* 1  ( xC* 2 )  xC* 2  ( xC* 2 )
C
*
C1
C
 ( x  x ) 
C
In
C -gange
x*
*
C1
*
C2
C
,
C
x  C ( x ) C ( xC* 2 )
*2
C2
*
C1
xC* 2 C ( xC* 2 )  xC* 1 C ( xC* 2 )
.
we can be wrote analogouse formulas if we make the changes


C
and
xC* .
With the help of these formulas are demonstrated the usual rules of arithmetic commutativity of
sum and product, associativity of sum and product, distributive rule.
Constructed rules permits calculation of opposite and inverse fuzzy numbers. For opposite fuzzy
nubmer one has:
~* ~* ~* ~*
a x 0
( 0  0).
Condition of existing opposit fuzzy number is:
Solution can be presented in the form:
~
  ( 0* )    (a~* ) .
~
x*   a * ;   ( ~
x * )    ( 0* )    (a~* ) .
For the inverse fuzzy number we can write:
~* ~* ~* ~*
a y 1
y* 
and
1
a*
( 1  1);
.
 
 .
~
a*  1 *    a~*
*
~
 y 
a *2
 
Estabhlised rules permit to recive solutions for algebric equations and the system of equations.
7 Conclusion
In this work is maked an attempt of use an quantum mechanical formalizm, which is universal
language of science in informatics.
Literature:
[1]
G.G.Emch : Algebraic Methods in Statistical Mechanics and Quantum Field Theory, John
Wiley and Sons , 1972.
[2]
D.Dubois and H.Prad : Theorie des possibilities , Musson , 1988.
[3]
P.A.M. Dirac : Principles of Quantum Mechanics, Clarendon Press, 1958;
[4]
S.R.de Groot, L.G.Suttorp : Foundation of Electrodynamics, North-Holland, 1972;
[5]
W.Heizenberg : Physical Principles of Quantum Theory, Clareddon Press , 1930.