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# G.Resconi INS-D-09-228R(5) 18 10 2013

Authors:
• Catholic University via Trieste 17 Brescia Italy

## Abstract

Given a n+1 dimensional space S with , we use the model ( hyper-plane) in S to control the transformation of the points into where the values y’ have the minimum distance from y. The algorithm to compute the parameters and the values y’ is denoted minimum action reasoning. The operation is the geometric projection of y into the hyper-plane in S. With the different models or hyper-planes we can control many different geometric transformations as reflection , rotation , refraction. With a chain of transformations we generate the minimum path in S that joins one point to another ( geodesic ). One ray in the space S is controlled by models as a special environment that guides the ray to have the task with minimum distance. The minimum action reasoning can be used to create software by models for different applications The coordinates of the space S can be real numbers , logic values , fuzzy sets or any other set that we can define. Classical linear or non-linear regression is part of this minimum action reasoning. Also classical logic, many value logic and fuzzy logic are included in the minimum action reasoning.
Control of the Minimum Action Reasoning
Soft computing by multi dimension optic geometry
GERMANO RESCONI
Faculty of Mathematics and Physics , Catholic University, Brescia , Italy
resconi@numerica.it
Abstract
Given a n+1 dimensional space S with
1 2
( , , ,...., )
n
y x x x S
, we use the model ( hyper-plane)
in S
1 1 1 2 1 2
( , ,...., ) ,..... ( , ,...., )
n q q n
y f x x x f x x x
β β
= + +
to control the transformation of the
points
{ }
1 1 2 2 1 2 1 2
( , , ,...., ), ( , , ,...., ),....,( , , ,...., )
n n k n
Y y x x x y x x x y x x x=
into
{ }
1 1 2 2 1 2 1 2
' ( ', , ,...., ),( ', , ,...., ),...., ( ', , ,...., )
n n k n
Y y x x x y x x x y x x x=
where the values y’ have
the minimum distance from y. The algorithm to compute the parameters
1 2
( , ,...., )
q
β β β
and
the values y’ is denoted minimum action reasoning. The operation is the geometric projection
of y into the hyper-plane in S. With the different models or hyper-planes we can control many
different geometric transformations as reflection , rotation , refraction. With a chain of
transformations we generate the minimum path in S that joins one point to another
( geodesic ). One ray in the space S is controlled by models as a special environment that guides
the ray to have the task with minimum distance. The minimum action reasoning can be used to
create software by models for different applications The coordinates of the space S can be real
numbers , logic values , fuzzy sets or any other set that we can define. Classical linear or non-
linear regression is part of this minimum action reasoning. Also classical logic, many value
logic and fuzzy logic are included in the minimum action reasoning.
Keywords
Minimum action, reasoning , geodesic, soft-computing, extension of linear regression , projection
operator , fuzzy number variables, geometry of fuzzy reasoning , reflection , rotation , refraction,
geometrical optics of logic.
1. Introduction
The paper studies the possibility to implement minimum action reasoning in a multidimensional
space S. The first step of the paper is to create an algorithm to project a random set of points y in S
into a set of points y’ that are in the best model within a family of models. The algorithm chooses
among all models in the family the best model for which y and y’ have the minimum distance. The
action to choose the best model is named minimum action (geodesic ). Any chain of minimum action
is denoted reasoning and therefore the algorithm is denoted minimum action reasoning. More
complex geometric transformations are possible as projection, reflection , rotation , refraction and so
on in the space S. For a family of linear models we can use the geometric projection to compute the
best linear parameters in the linear regression. We can also extend the linear regression to non -
linear regression or to fuzzy number transformations. The space S can be a space of logic values , as
classical logic values , many value logic and fuzzy logic.
1
2 Linear regression ,projection operator, minimum action reasoning
Specialized literature on regression analysis (Gujarati 2003 [2]) and more generally on linear and non
linear models (Ryan 1997 [2] ) offered many solutions to study the dependence between two types of
variables y and {x1,x2,...,xp } where y is a quantitative dependent variable and {x1,x2,...,xp } are
independent variables. Regression analysis studies the dependence of y with respect to the variables
{x1,x2,...,xp }when we have samples of y and samples of {x1,x2,...,xp }. This requires the choice of a
suitable model and the related parameters estimation. Given the generic model:
y = f ( x1,...,xp ; β )+ε (1)
the statistical regression aims to find the set of unknown parameters so that
%
°
( , ,..., ; )
1 2
yf x x x p
β
=
(2)
Where
%
y
are the values of y in agreement with the model and with the minimum errors respect to the
given samples of y. The term ε indicates the deviation of y from the model. The most widely used
regression model is the Multiple Linear Regression Model (MLRM), as well as the Least Squares
(LS) is the most widespread estimation procedure. In the MLRM the dependent variable y would be
expressed as the weighted sum of the independent variables {x1,x2,...,xp}, with the unknown
parameters
12,...,βp }
Formally we have the hyper-plane
......... ,
0 1 ,1 p
y x x
n n p n
n
β β β ε
= + + + +
(3)
where β0 is the parameter related to intercept term. In a matrix form the model is expressed as:
y=X β+ε
where
(4)
LS is based on the minimization of the sum of squared deviations:
min ( ) ( )
T
D y X y X
T
where () is the matrix transpose
β β
β
= −
(5)
The optimal solution β of the minimization problem is obtained in this way
2
( ) ( )
( ) ( ) ( )
T
D y X y X
T T T T
y y y X X y X X
β β
β β β β
= − =
− − +
To compute the minimum value we make the derivatives of the previous form
T T
DT T T T T
y X X y X X X X
j j j j j
β β β β
β β
β β β β β
∂ ∂
= − + +
∂ ∂
where
10
... ...
1
and , 0 ... 1 0 ... 0
0
j j
1
...
...
0
p
T
jT
v v
j j
j
β
ββ β
ββ β
β
β
   
   
   
   
     
   
 
   
   
   
   
 
 
 
∂ ∂
= = = = =
∂ ∂
+
We have
0
D
j
T T T T T T T
for y Xv v X y v X X X Xv
j j j j
β
β β
=
+ = +
But because we have the following scalar property
( )
T T T T
P A B A B B A= = =
the previous expression can be written as follows :
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
T
T T T T T
T T
v X y v X y X y v y Xv
jj j
j
T
T T T T T T T T T
T T
v X X v X X X X v X X v X X v
jj j j j j
j
β β β β β
 
 
 
 
 
 
= = =
= = = =
We have
2
2
2 2
T T T T T
y Xv v X y v X y
j j j
T T T T T T
v X X X Xv v X X
j j j
and
T T T T
v X y v X X
j j
β β β
β
+ =
+ =
=
3
whose solution is
1
( )
T T
X y X X
T T
X X X y
β
β
=
=
(6)
For the previous solution for the optimal condition we obtain
1
( )
T T
y X X X X y Qy
ε ε
= + = +
(7)
We remark that the operator
1
( )
T T
Q X X X X
=
is a projection operator for which
2 1 1 1
( ) ( ) ( )
T T T T T T
Q X X X X X X X X X X X X Q
− −
= = =
Geometric image of the projection operator
Figure 1 Projection of the vector y into the plane of two dimensions X = ( x1 , x2 ).
Example 1
1 1 1
1 2 , 2
1 3 2
X y
 
 
= =
 
 
 
Parameters
2
3
1
2
T -1 T
= (X X) X y
β
=
 
 
 
 
 
Projection vector
(1- Q)y
Qy = z
x1
x2
s 1
s 2
s 3 y
4
3 1(1) 2
3 2 2
1 1 3 1 5
12
( ) 1 2 (2)
1 2 2 2
1 3 3 1 3
2(3)
2 2
T T
Qy X X X X y
 
+
 
 
   
 
   
= = = + =
 
   
 
   
   
 
   
 
+
 
The values in the projection vector are samples of the best fit of this linear form
Graph in figure 2 with points in the vector y with the previous linear model
0 1 2 3 4
0.5
1
1.5
2
2.5
3
2.667
0.667
R
h 1,
Y
k
40 R
h 0,
t
k
,
Figure 2 Best fit linear form with original point y.
The projection of y into the plane X is the minimum path for which the three values in y ( cluster of
points ) move to the final point on the straight line. Now for
1
( )
( )
T T
z X X X X y Qy
and
p I Q y
= =
= −
We have that z is orthogonal to y in fact
5
1 2
2 3
y x
= +
( ) [( ) ] ( )
1 1
( ( ) ) ( )
2
( ) ( ) 0
T T T T
z p QY I Q Y Y Q I Q Y
but
T T T T T T
Q X X X X X X X X Q
So
T
Q I Q Q I Q Q Q Q Q
= − =
− −
= = =
= = − = − =
Now the projection is a segment line from y to its projection Q y which is on a straight line
( minimum distance ) . The movement from y to Q y is the minimum action whose points are
, 1
1,
,
p
y z where k
kk
when
k y z p y
k
when
k y QY z
k
= +
= = + =
→ ∞ = =
Graphic image for k =1 , k=2 , k=3, k=4
Figure 3 The set of points represented by squares are the initial values y. The black points are
minimum action from y to Q y which are represented by rhombus.
The algorithm by which we move from the initial points y to the projection Q y with the inter-media
values ( movement ) is denoted minimum action reasoning.
6
Figure 4 When we change the initial points y , the movement of the points changes as we can see in
this figure.
We show in figure 5 the projection Q y , and the movement on the segment line ( 1 – Q ) y by which
we move from y to Q y.
Figure 5 The minimum action reasoning movement from y to the projection Q y by a straight line or
ray
Example 2
Given the non - linear family of models
( ) ( )
1 1 2 2
2
( ) 1, ( ) ( 4 2)
1 2
y f x f x
f x f x x x
β β
= +
= = − +
we compute the basis vectors : one for
0
2
β
=
and the other for
0
1
β
=
so we have the two
dimensional plane
(1- Q)y
Qy = z
X 1
X2
S 1
S 2
S 3 y
Minimum action
reasoning path
7
( ) ( )
1 2
1 1 1
2 1 2
3 1 1
x f x f x
x
Xx
x
=
==
=
 
 
 
 
 
 
In a short form
1 1
1 2
1 1
X
 
 
= 
 
 
For the random value of y given by the vector
1
2
2
y
 
 
= 
 
 
we have the parameters
1
1
2
T -1 T
= (X X) X y
β
= 
 
 
 
and the projection operator
1
1 (1) 3
22
1
y 1 (2) 2
23
1
1 (1) 2
2
T -1 T
z X = X(X X) X y Q
β
 
+ 
 
 
 
= = = + =  
 
 
 
+ 
 
The vector y is a linear combination of the colon vectors in X.
1
2 2
( 4 2) 1 ( 4 2)
1 2 2
y x x x x
β β
= + − + = + − +
2
2
2
1
1 ( 1 4 2) 3
2
(1) 1.5
2
1
(2) 1 ( 2 4(2) 2) 2 2
2
(3) 3 1.5
1
1 ( 3 4(3) 2) 2
2
y
y y
y
 
+ + −  
 
 
 
 
 
= = + − + = =
 
 
 
 
 
 
 
 
+ − +  
 
 
8
1 1 1
1
1
Because
( ( ) ) ( ( ( ) ) ) ( ( ) )
( ( ) ) 0
so
( ( ) ) is orthogonal to y
T T T T T T T T
T T
T T
I y y y y y I y y y y y I y y y y y
and
I y y y y y
I y y y y
− − −
= − = −
− =
Numerically we have the three vectors orthogonal to y
25 6 9
34 17 34
6 9 6
1
( ( ) ) 1 2 3
17 17 17
9 6 25
34 17 34
T T
p I y y y y p p p
− −
= = − =
− −
 
 
 
 
 
 
 
 
The previous column vectors are orthogonal to y. In a graphic way we have
.
Figure 6 The tree vectors
1
( ( ) )
T T
I y y y y
orthogonal to the vector y
2
,
we have
[( ) ] ( ) 0
because
( ) 0
and
[( ) ] 0
Because
T
Q Q Q Q
T T T
I Q y Qy y I Q Qy
T
I Q Q
T
I Q y Qy
= =
= − =
− =
− =
9
y = (1.5,2,1.5)
p2
p1
p3
So ( I – Q ) y is orthogonal to Q y. Numerically we have
1
2
[( ) ] 0
1
2
I Q y
 
 
 
=  
 
 
 
In a graphic way we have
Figure 7 We show in the three white triangles the input data y ; the black points are Q y , the white
squares are the points ( I – Q) y orthogonal to the black points Q y.
For the (7) we obtain
01234
1
0
1
2
3
X
k
D
k
C
k
F V
h
( )
k 1
+
k 1
+,
k 1
+,
V
h
,
10
1
( ) (1 )
( ) ( ) (1 )
2
(1 ) (1 ) ( ) 0
T T
y X X X X y y Qy Q y
and
T T
Qy Qy Q y
T T T T
y Q Q y y Q Q y y Q Q y
ε
ε
= − = =
= − =
− = − = =
The error ε is perpendicular to the optimal condition Q y. We remark also that
( ) (1 ) 0Q y Qy Q Q y Q
ε
= = =
The projection of the error is equal to zero and therefore
1
( ) ( ) ( )
T T
Q y X X X X y Qy Q Qy
ε ε ε
+ = + = + =
(8)
The projection operator Q projects the variable y + ε into Q y where the error is eliminated.
3. Metric G in the parameter space
Let’s prove that the minimum action reasoning can be written as follows :
min ( )
y X
T T T
P X X G
T
E X y
β
β β β β
β
=
= =
=
(8)
where P is the minimum distance ( geodesic ) in the space of the parameters β with a metric G. The
form E = XT y is the constrain in the minimum action reasoning that is invariant for the projection
operator Q. In fact we have
XT Q y = XT y = E (9)
Proof :
To solve the minimum problem with constrains we use the Lagrange multipliers and thus
( )
( ) ( )
T T
D G E X y
T T T
G E X X G E G
β β λ
β β λ β β β λ β
= + − =
+ − = + −
Now we compute the derivative in this way
0
T
DT
G G G
j j j j
β β β
β β λ
β β β β
∂ ∂
= + − =
∂ ∂
that can be written as follows
11
2
T T T
v G Gv Gv Gv
j j j j
β β β λ
+ = =
For which
2T
λ β
=
and
( ) 2 ( )
2 2 2
T T
D G E G
T T T T T
G E G E G
β β β β β
β β β β β β β β
= + − =
+ − = −
We have also that
2( ) ( ) ( ) ( ) ( )
2 ( )
T T T T T
y X y X y y y X X y X X
T T T T T T T T
y y E E G y y E G y y D
ε β β β β β β
β β β β β β β β
= − = +
= − + = − + = −
If D(β) assumes the minimum value also the error ε2 assumes the minimum value.
Example
2 2
2 2 (2 2 2 )
1 1 2 1 2
1 2 2
2( 2 ) 0
1 2
1
1
2( 2 ) 0
2 1
2
2
D E E
DE
DE
β β β β β β
β β
β
β β
β
= + + +
= − − =
= − =
So we have
21 2
1
22 1
2
E
E
β β
β β
= +
= +
or
1 0 1 0 2
1 1 1 2
( 1 1 1 1 ) 2
2 2 2 1
0 1 0 1
1
,
1
T
E
E
or
E G G E
T
y X XG X Y QY
β β β
β β β
β β
β
+
= = +
= =
= = =
   
 
   
 
   
 
   
In a general case we have that the minimum condition is
12
2 ( )
2 2 0
T T
DT
E G G
j j j j
T T
v E v G
j j
β β β
β β
β β β β
β
∂ ∂
= − + =
∂ ∂
− =
The solution is
E G
β
=
for which we have
1 1 T
G E G X y
β
− −
= =
(10)
And for the minimum action reasoning for β we have the projection operator
1
T
y X X G X y Qy
β
= = =
(11)
The (11) is the minimum action reasoning by projection operator. Now for the error ε, for which
Q ε = 0, we obtain
1
( )
T
y X X G X y
Qy Q Qy
β ε
ε
= = + =
+ =
(12)
The projection operator separates the variable y from its error ε. The elimination of the error ε from
the original variable y in the projection operation gives the meaning of the optimal condition for β.
We remark that the minimum action reasoning is generated by a conditional minimum with
constrain (8) without the computation of the variance.
Example 3
Given the column space
1 0
1 1
0 1
X
 
 
= 
 
 
We have
13
1 0
1 1 1
1
1 1
2 2 1 2
2
0 1
3 3 2
1 0 1 0 1 0 1 0
1 1 1 1
1 1 1 1 ( 1 1 ) 1 1
2 2 2 2
0 1 0 1 0 1 0 1
2
( ) ( ) 1 2
q y
q Q y
q y
T
T
T
P
T
Qy Qy q q
β
ββ β
ββ
β β β β
β β β β
   
   
= = = +
   
   
 
 
= =
= = +
  
  
  
 
       
       
       
       
       
       
       
2 2 2 2 2 2 2
( ) 2 2 2
1 1 2 2 1 2 1 2
3
q
β β β β β β β β
+ = + + + = + +
So in the space of the samples q the form P is a simple quadratic form and the geometry is the
traditional Euclidean space . In the space of the parameters β the form P is a quadratic form with a
cross ( non –Euclidean space ) term that gives the dependence between the two vectors
1 0
1 , 1
1 2
0 1
x x= =
 
 
 
 
In a graphic way we have
Figure 8 Non orthogonal reference space in the two dimensional plane generated by the vectors x1 ,
x2 as non Euclidean space. The space ( q1 , q2 ,q3 ) is the sample space.
We remark that the unitary transformation U for which we have
1
( ) ( )
T
U U
and
T T T T
P U UG U UG G
β β β β β β
=
= = =
(13)
is a transformation for which P is invariant. When P assumes the minimum value, the change of the
parameters β and Gβ by U does not change the minimum value P. The transformation U is the
unitary transformation that gives all the possible parameters for which we have the minimum value
for P. This is similar to the least action in physics. Any change of the reference does not change the
least action property in mechanics.
x 1
x2
q 1
q 2
q3
14
4. Minimum action reasoning in the fuzzy number space.
In this chapter we suggest a new representation of the classical fuzzy inference process [1] ,[2],[3] by
extension of the linear regression to minimum action reasoning by projection operators and to
variables or coordinates whose values are fuzzy numbers.
Let
1 2
( , , ,...., )
n
y x x x S
where
( , , ,...., ) .....
1 2 1
y x x x S U U U
n y x xn
∈ =
Each universe Uj is a domain whose values are fuzzy numbers. The fuzzy model is
1 1 1 2 1 2
( , ,...., ) ,..... ( , ,...., )
n q q n
y f x x x f x x x
β β
= + +
where the basis functions are functions whose independent variables and dependent variables are
fuzzy numbers.
1 1 2 1
1 1 2 1
1 2
( , ,...., )
( , ,...., )
.........................................
( , ,...., )
n
n
q n q
f x x x y U
f x x x y U
f x x x y U
= ∈
= ∈
= ∈
Where U is a collection of fuzzy numbers. In the minimum action reasoning in the real numbers we
have
1 ...
1,1 1,
1
1 ...
2,1 2,
2,
... ... ... ... ...
1 ...
,1 ,
q
x x p
y
x x
yp
y X
yx x
q q p
 
   
   
   
   
   
   
 
   
= =
When we substitute the ordinary numbers with the fuzzy numbers
,i j
A
we obtain for X the matrix
( ) ( ) ... ( )
1,1 1,2 1,
( ) ( ) ... ( )
2,1 2,2 2,
( )
... ... ... ...
( ) ( ) ... ( )
,
,1 ,2
A x A x A x
p
A x A x A x
p
X x
A x A x A x
q p
q q
 
 
 
 
= 
 
 
 
where
X
is the fuzzy connection matrix. We have also that for the random fuzzy number
( )
1
( )
2
( ) ...
( )
B y
B y
Y y
B y
q
 
 
 
= 
 
 
 
15
In the same way in which we have computed the parameters β in chapter 2 , we can compute the
functional parameters β in the functional space of the fuzzy numbers
1
( , ) ( ( ) ( )) ( ) ( )
T T
x y X x X x X x Y y
β
=
(14)
Where
2( ) ( ) ( ) ... ( ) ( )
,1 ,1 ,2 ,1 ,
2
( ) ( ) ( ) ... ( ) ( )
,2 ,1 ,2 ,2 ,
( ) ( ( ) ( ))
... ... ... ...
2
( ) ( ) ( ) ( ) ... ( )
, ,1 , ,2 ,
A x A x A x A x A x
j j j j j n
j j j
A x A x A x A x A x
j j j j j n
T
G x X x X x j j j
A x A x A x A x A x
j n j j n j j n
j j j
 
∑ ∑
 
 
 
∑ ∑
 
= =  
 
 
 
∑ ∑
 
 
Given the functions (14) we can build the best output of fuzzy numbers with the expression
1
( , ) ( ) ( , ) ( )( ( ) ( )) ( ) ( )
T T
Y x y X x x y X x X x X x X x Y y
β
= =
where
( , ) ( ) ( )
T
E x y X x Y y=
is the functional input that is connected with the coefficients (14) in the following way
1 1
( , ) ( ( ) ( )) ( , ) ( ) ( , )
T
x y X x X x E x y G x E x y
β
− −
= =
The functions
( , ), ( , )x y E x y
β
are dual functions for which we have
1
( , ) ( , ) ( ) ( , ) ( , ) ( ) ( , )
T T
P x y x y G x x y E x y G x E x y
β β
= =
And the (8) is
( , ) ( ) ( , ) ( , ) ( )
min ( , ) ( ( ) ( )) ( )
( , ) ( ) ( )
Y x y X x x y Q x y Y y
T T T
P x y X x X x G x
T
E x y X x Y y
β
β β β β
β
= =
= =
=
Now given
( , )x y
β
we have a fuzzy model by which given a set of fuzzy sets as values for the input
variables
( , ,...., )
1 2
x x xn
we can compute the associate fuzzy set for the output variable y. In fact we
have
16
( , ) ( ) ( , ) ( ) ........ ( , ) ( )
1 1
Q y x B y x y A x x y A x
p p
β β
= + +
5. Fuzzy logic and classical logic by projection operator
5.1 Classical Logic Models
For classical logic the AND , OR , IF...THEN , NOT ..... rules can be represented by the model
( , ) 1 2 3 4
z x y x y xy
β β β β
= + + +
In the numerical way we have
11
0 0 0 1 0
,
0 1 0 1 0
1 0 0 1 1
1 1 1 1
x y xy
X y
 
 
 
 
= =  
 
 
 
 
For chapter 2 we have
1
21
3
4
1
1
( ) 2
1
T T
X X X y
ββ
βββ
   
   
   
= = =
   
   
 
 
The projection operator is
0 0 0 1 1 1
0 1 0 1 1 0
1 0 0 1 2 0
1 1 1 1 1 1
2
1 ( 2 ) 1 ( )
z Qy X
z x y xy x y
β
= = = =
= − + = −
 
 
 
 
 
 
For the logic operation y = A B “if A then B” ,we have y(0,0) = 1 , y(0,1) = 1 , y(1,0) = 0 and
y(1,1) =1 . So
17
0 0 0 1 1
0 1 0 1 1
,
1 0 0 1 0
1 1 1 1 1
X y
 
 
 
= =
 
 
 
And the parameters are
1
21
3
4
1
0
( ) 1
1
T T
X X X y
ββ
βββ
   
   
   
= = =
   
   
 
 
So the model of inferential rule “if A then B” is the following
1 ( 1) 1z x xy x y= − + + = +
5.2 Many Value Logic and Fuzzy Logic models
Given the possible values
1 2 1 2
( ) , ( )
2 2
x x y y
x y
µ µ
+ +
= =
where x and y are classical logic values one and zero ,we obtain the many value logic
1 1
( ) 0, ,1 , ( ) 0, ,1
2 2
x y
µ µ
   
∈ ∈
   
   
The general composition rule is
0 1 1 2 2 3 1 4 2 5 1 1 6 1 2 7 2 1 8 2 2
z x x y y x y x y x y x y
β β β β β β β β β
= + + + + + + + +
We can simplify and generalise the previous model in this way
3
2 4
1
3 4
1 2
z x y x y
j j j j
n n n
j j j
or
z x y xy
β
β β
β
β β β β
= + + +
∑ ∑
= + + +
where is the average value
Example 4
18
For the AND operation in the classical logic we have
1
0, 0, 0,
1 2 3 4 2
β β β β
= = = =
In many value model we have
1 1 2 2
( )
2
x y x y
z+
=
So we have the composition rule
1 1 2 2
1 1
0 1
2 2 2
0 0 0 0 0
1 1 1 1
0
( ) 2 2 2 2
1 1 1
0 0
2 2 2
1 1
1 0 1
2 2
x y x y
z
p q
µ
+
 
=
 
 
 
 
 
=  
 
 
 
 
 
 
that can be separate in two AND operations
1 1
0 1 0 1
2 2
0 0 0 0 0 0 0 0
min( , ) , min( , )
1 1 1 1 1
0 0 0
2 2 2 2 2
1 1
1 0 1 1 0 1
2 2
x y x y
q x y q x y
   
∧ ∧
   
   
   
   
= = = ≤
   
   
   
   
   
6. Minimal action reasoning for fuzzy sets
Given the triangular basis of fuzzy set
(1 )
( ) ( )
0
x u
i j x u u
i j
u
A x x
j i
j
x u u
i j
µ
≤ ∆
= = > ∆
19
in a graphic way we have
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1
0
M
h 0
,
M
h 1
,
M
h 2
,
M
h 3
,
M
h 4
,
20 x
h
Figure 9 Set of triangular fuzzy set
We have the connection matrix
( )
1
( )
2
( ) ...
( )
5
A x
A x
X x
A x
 
 
 
= 
 
 
 
In a numerical way we have the five fuzzy sets A , one for each row. The five fuzzy sets are
presented in the matrix A, one for each column
A
0.1
0.2
0.6
1
0.6
0.2
0.1
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0.1
0.2
0.6
1
0.6
0.2
0.1
:=
Now given the trapezoid fuzzy set Y(x) whose values are Yp in figure 10
20
0 5 10
0
2
4
3
0
Y
p
100 p
Figure 10 Trapezoid fuzzy set
in a numerical way we have
0
1
2
3
3
3
3
2
1
0
0
Y
 
 
 
 
 
 
 
 
= 
 
 
 
 
 
 
 
 
With the samples A of the five fuzzy sets and the input trapezoid fuzzy set Y , we can project Y into
the five sets as a five dimension plane embedded in the 11 sample space. So we have
1
0.303
0.605
1.785
3.232 3.026
2.911 0.151
( )( ( ) ( )) ( ) ( ) , ( , )
2.973 1.482
3.044 1.307
2.039 0.958
0.836
0.192
0.096
T T
QY X x X x X x X x Y y x y
β
 
 
 
 
   
   
 
 
   
= = =
   
   
   
   
 
 
 
 
 
21
In a graphic way QY ,whose numerical values are QAp in figure 11
0 5 10
0
2
4
3.232
0.096
QA
p
100 p
Figure 11 Projection of the trapezoid fuzzy set into the subspace of the fuzzy sets A
and QY = 3.019 A1 -0151 A2 + 1.482 A3 + 1.307 A4 + 0.958 A5 which is the best Transformation of
the trapezoid Y into Y’ that is the linear composition of the original basis fuzzy sets The parameters
β show the weight order of the fuzzy sets A given by the trapezoid input , for which we have
13.019
1.482
3
( ) 1.307
4
0.958
50151
2
A
A
A
order A
A
A
 
 
 
 
= =
 
 
 
 
 
7. Minimal action reasoning by oblique projection
In a graphic way we have the oblique projection
Figure 12 Oblique projection of y into B through A.
22
y
A
B
Y = B ( AT B )-1 AT y
K = A ( AT A )-1 AT y
Given the projection operator
1
( )
T T
Q y A A A A y
=
we see that the oblique projection operator P is a vector whose projection on A is Q y . For this
remark we can compute the form of the projection operator in this way
1
1
T -1 T T -1 T
Q y = A(A A) A y = A(A A) A P y
T -1 T T T T -1 T
= A(A A) A [B(A B) A ]y = A(A A) A y
so
T T
P = B(A B) A
And P is the oblique projection .
Example 5
Given the plane in three dimensions ( colon space )
1 1
1 2
1 3
A
 
 
= 
 
 
The sample for y is
1
1 2
2
y
 
 
= 
 
 
The orthogonal projection is
1
1.167
( ) 1 1.667
2.167
T T
y A A A A y
 
 
= =  
 
 
Now for B = Z A we have the oblique projection
1 1
2 ( ) 1 ( ) 1
T T T T
y B A B A y ZA A ZA A y
− −
= =
So the orthogonal projection of z on the plane X is equal to the projection of y1 on the same plane
X. For
1 0 0
0 3 0
0 0 1
Z
 
 
= 
 
 
we have
23
0.714
2 2.571
1.714
y
 
 
= 
 
 
In figure 18 we see the transformation from y1 into YR = Q y1 and also into P y = y2
Figure 13 The original samples y1 are the rhombus points ; the model is the straight line , the points
on the straight line are the projections of the rhombus points on the straight line with the minimum
error. The square are the oblique projection of the samples y1
Example 6
We show the fuzzy sets A and the fuzzy sets B in this way
A
0.1
0.2
0.6
1
0.6
0.2
0.1
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0.1
0.2
0.6
1
0.6
0.2
0.1
:= B
0.1
0.2
0.6
0.6
0.6
0.2
0.1
0
0
0
0
0
0
0.2
0.6
0.6
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
0.6
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
0.6
0.6
0.2
0
0
0
0
0
0
0.1
0.2
0.6
0.6
0.6
0.2
0.1
:=
In a graphic way the basis fuzzy sets A are
24
0 2 4 6 8 10
0
0.5
1
1
0
A
k 0
,
A
k 1
,
A
k 2
,
A
k 3
,
A
k 4
,
100 k
Figure 14 Set of fuzzy sets A
the basis of the fuzzy sets B = Z A is
0 2 4 6 8 10
0
0.2
0.4
0.6
0.7
0
B
k 0
,
B
k 1
,
B
k 2
,
B
k 3
,
B
k 4
,
100 k
Figure 15 Fuzzy sets B
By oblique projection from the trapezoid distribution yp we have QAp by the orthogonal projection
and after by oblique projection Q ( see figure 12 ) we obtain Qyp as we can see in figure 16.
0 5 10
0
2
4
y
p
p
0 5 10
0
2
4
QA
p
p
0 5 10
2
0
2
4
Qy
p
p
A B
y
Figure 16 Projection from y to B by A by oblique projection operator.
25
8. Metric G for mixed spaces A and B in minimal action reasoning
For the oblique operator where B = Z A we have the minimal condition
( )
min ( )
y B Z A
T T T
P A B G
T
E A y
β β
β β β β
β
= =
= =
=
(15)
where the metric of the parameter space is
T
G A B=
and the transformation from y to y’ is the
oblique projection
Proof :
To prove (15) we can repeat the same computation in (8) by Lagrange multipliers and obtain
( )
( )
T T
D G E A B
TG E G
β β λ β
β β λ β
== + − =
+ −
So
0 for =2
D
j
λ β
β
=
2 ( )
0
T T
D G E G
D for E = G β
j
β β β β
β
= +
=
We remark that for the constraint E = AT y , we have the same property of the classical linear
regression with
E = AT y = (AT B) ( AT B )-1 AT y = AT ( B (AT B )-1 AT y ) = AT Q y
In conclusion we have the new type of projection operator to compute the parameters of the model
1
T
y B B G A y Qy
β
= = =
(16)
where Q is a projection operator. In fact
-1 T T -1 T
Q = B G A = B (A B) A
2 T -1 T T -1 T T -1 T
Q = B (A B) A B (A B) A = B (A B) A
(17)
26
9. Optical Geometry of Fuzzy reasoning
9.1 Reflection by projection operator
Now we know that the reflection operator is represented by the graph
Figure 17 Reflection of y in Ref y
As we can see in figure 22 we have
( )
( ) (2 )
y Qy I Q y
Ref y = Qy I Q y Qy Qy y Q I y
= + −
= + − =
The reflection point of y is function of the projection operator Q.
We remark that
Ref(Ref (y )= (2Q - I)(2Q - 1)y = (4Q - 2Q - 2Q + I)y = y
and
QRef(y)= Q(2Q - I)y = (2Q - Q)y = Qy
Example 7
Given the plane in three dimensions ( colon space )
1 1
1 2
1 3
A
 
 
= 
 
 
given the sample
O
y
Ref y
K
A
Q y
(I- Q )y
-(I- Q )y
27
1
1 2
2
y
 
 
= 
 
 
for the reflection operator we have
4
3
4
1
(2 ) 1 (2 ( ) ) 1 3
7
3
T T
Ref y1 = Q I y A A A A I y
 
 
 
 
− = − =  
 
 
 
Figure 18 the rhombus points are the original points , the dot points are the projection points and the
squares are the points generated by the reflection operator.
Example 8
For the two dimensional space of samples we have
cos( ) cos( )
,
sin( ) sin( )
y A
α β
α β
 
= =
 
 
28
1
( ) ( )( ( ) ( )) ( )
sin(2 )
2
cos( )
cos( ) cos( ) cos( ) cos( )
12
( )
sin( ) sin( ) sin( ) sin( ) sin(2 ) 2
sin( )
2
T T
Q A A A A
T T
β β β β β
β
β
β β β β
β β β β β β
=
= =
 
 
         
       
         
 
And the reflection operator is
cos(2 ) sin(2 )
( ) sin(2 ) cos(2 )
Ref( )= 2Q I
β β
β β β β
 
=  
 
Given y , we have the reflection
( ) ) ( )
cos(2 ) sin(2 ) cos( ) cos( )cos(2 ) sin( )sin(2 )
sin(2 ) cos(2 ) sin( ) cos( )sin(2 ) sin( ) cos(2 )
Ref( )y( ) = (2Q I y
β α β α
β β α α β α β
β β α α β α β
+
 
= =
 
− −
 
Numerical example 9
= 60, = 45
α β
We have
3
cos(2(45)) sin(2(45)) cos(60) cos(30)
2
sin(2(45)) cos(2(45)) sin(60) sin(30)
1
2
Ref( )y( )
β α
 
 
   
 = = =
   
 
   
 
 
9.2 Minimum path and reflections
Now given two points x and y , we want to find the minimum path between x and y that is reflected
in C by the model ( hyper-plane in S )
Figure 19 Reflection from x to y by C in A
x
y
Ref y
C
A
29
Now given the points
cos( ) cos( ) cos( )
, ,
sin( ) sin( ) sin( )
x y A
α ρ β γ
α ρ β γ
 
= = =
 
 
we compute the reflection point C
(2 )
cos(2 ) sin(2 ) cos( ) cos(2 ) cos( ) sin(2 )sin( )
sin(2 ) cos(2 ) sin( ) sin(2 ) cos( ) cos(2 )sin( )
Ref y Q I y
γ γ ρ β γ β γ β
ρ
γ γ ρ β γ β γ β
= − +
 
= =
 
− −
 
With the graph
Figure 20 Oblique projection and reflection
Now we compute the vector F whose origin is in x and the end in P x
(2 )
cos( ) cos(2 ) cos( ) sin(2 )sin( )
sin( ) sin(2 )cos( ) cos(2 )sin( )
F = x - Ref y x Q I y
α δ β δ β
ρ
α δ β δ β
= − +
 
= −
 
 
The vector orthogonal to the vector F is
1
( )
T T
E I F F F F
= −
In fact we have
1
( ( ) ) 0
T T
EF I F F F F F
= − =
30
x
y
Ref y
P x = C
Column space A
constrain
E
Oblique
projections
F
With the expression of F we have
2 2 2
2
2
2 sin( ) 2 sin( ) sin( ) sin( 2 )
2 2
4sin( ) 2 1
2
[cos( ) cos( 2 )][(sin( ) sin( 2 )]
2cos( 2 ) 1
R R R
R R
E
R R
R R
α β α β
δ δ α β δ
α β δ
α β δ α β δ
α β δ
+ −
 
− − − + +
 
 
+
+ − − +
= 
 
+ −
 
+ + − +
 
The point C is given by the oblique projection whose operator was computed in the previous chapter
1
( )
T T
C A E A E
=
or
2
cos( )[2cos(2 2 ) 2 cos( 2 ) 2 cos( 2 4 ) 2 cos( )]
sin( )[2 cos(2 2 ) 2 cos( 2 ) 2 cos( 2 4 ) 2 cos( )]
where
2[cos(2 ) cos( ) cos(2 3 )
cos( ) 2 cos( )
R R R
Det
CR R R
Det
Det R
R R
γ α β γ β γ α β γ α
γ α β γ β γ α β γ α
α γ γ β γ
α β γ α β γ
+ − − − + + − −
 
 
= 
+ − − − + + − −
 
 
 
= − − +
+ + − + +
2
cos( 3 ) cos( )R R
α β γ γ
+ −
Example 10
Given
2 1
2 , 2
1 2
x y
 
 
= =
 
 
 
and
1 1
1 2
1 3
A
 
 
= 
 
 
Now we want to compute the minimum action from x to y by C. Before we use the reflection
operator to obtain ref (y) see figure 25.
31
4
3
4
1
(2 ) (2 ( ) ) 3
7
3
T T
Ref y = Q I y A A A A I y
 
 
 
 
− = − =  
 
 
 
after we compute the segment that joins x with the ref ( y ).
2
3
2
3
4
3
x - Ref y
 
 
 
 
= 
 
 
 
after we compute the vectors E
5 1 1
6 6 3
1 5 1
1
)6 6 3
1 1 1
3 3 3
T
T
E = I - (x - Ref y)((x - Ref y) (x - Ref y) (x - Ref y)
 
 
 
 
= −
 
 
 
 
Because the determinant of E is equal to zero we choose only two colons that give us the plane
perpendicular to ( x – ref y ). So we have
5 1
6 6
1 5
6 6
1 1
3 3
E
 
 
 
 
= −
 
 
 
 
Now we are ready to project in an oblique way the three points x into the plane A to have the point
C. So we have
5
3
5
1
3
5
3
T
C = A(E A) Ex
 
 
 
 
= 
 
 
 
where C is the wanted result. We show in 26 the vectors x , C , y.
32
Figure 21 Given the points x and y we compute the point C for which from x we can generate y as
we can see in figure 20
9.3 Rotation by reflection and projection
We know that the rotation is the composition of two reflections as follows
Ref( )Ref( ) = Rot(2( - ))
β α β α
When we establish this angle of rotation
β γ
we have
= 2( - ))
β γ β α
So
2
2 2
and
β γ β α
β γ β γ
α β
= −
− +
= − =
We can decompose the rotation into two reflections. In fact we have
cos( ) sin( )
( )
( ) sin( ) cos( )
2
Ref( ) Ref
β γ β γ
β γ
αβ γ β γ
+ +
 
+
= =  
+ − +
 
And now we have
33
2
cos(2 ) sin(2 ) cos( ) sin( )
sin(2 ) cos(2 ) sin( ) cos( )
cos( ) sin( ) ( )
sin( ) cos( )
Ref( )Ref( )=
=
Rot
β γ
β
β β β γ β γ
β β β γ β γ
β γ β γ β γ
β γ β γ
+
+ +
 
 
+ − +
 
− −
 
= = −
 
− −
 
In a graphic way we have
Figure 22 Rotation by reflection
Remark
Each rotation can be written as a composition of projection operators
In fact we have
)
2 2
42 2
cos(2 ) sin(2 ) cos( ) sin( )
sin(2 ) cos(2 ) sin( ) cos( )
cos( ) sin( ) ( )
sin( ) cos( )
Ref( )Ref( ) = (2Q( ) - I)(2Q( ) I
I Q( )Q( ) 2Q( ) 2Q( )
=
Rot
β γ β γ
β β
β γ β γ
β β
β β β γ β γ
β β β γ β γ
β γ β γ β γ
β γ β γ
+ +
+ +
= +
+ +
 
 
+ − +
 
− −
 
= = −
 
− −
 
γ
β
2
β γ
+
34
With reflection and rotation we have all possible cases
All these operators can be decomposed into a chain of projection operators.
Because each complex rotation or orthogonal matrix in n dimensions can be decomposed in this way
1 2 1 2n n
Rot(θ , ,... )= Rot(θ )Rot( ),...Rot( )
θ θ θ θ
and because each rotation can be represented by two reflections we can decompose rach rotation into
2n reflections.
Example 11
In three dimensions we have
1 1
1 1 1
2 1
2
1 1
3 3 3
3 3
cos( ) sin( ) 0
sin( ) cos( ) 0
0 0 1
cos( ) 0 sin( )
0 1 0
sin( ) 0 cos( )
1 0 0
0 cos( ) sin( )
0 sin( ) cos( )
Rot(θ )=
Rot( )=
Rot( )=
θ θ
θ θ
θ θ
θθ θ
θ θ θ
θ θ
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Each rotation can be decomposed into two reflections and four projection operators. Each three
dimensions complex rotation can be decomposed into 6 reflections and 6 projections.
9.4 Refraction as composition of projection operators
With
cos( ) cos( ) cos( )
, ,
sin( ) sin( ) sin( )
x A B
α β γ
α β γ
   
= = =
   
   
the first projection from x to y into the space A is
35
sin(2 )
2
cos( )
cos( ) cos( ) cos( ) cos( )
12
( )
sin( ) sin( ) sin( ) sin( ) sin(2 ) 2
sin( )
2
T T
QA
β
β
β β β β
β β β β β β
= =
 
 
         
       
         
 
So we have
sin(2 )
2
cos( ) 2
sin(2 ) 2
sin( )
2
cos( )
sin( )
A
y Q x
β
β
ββ
α
α
 
  
= =   
 
 
 
And with
2
sin( )
2
cos( )
cos( ) cos( ) cos( ) cos( )
12
( ) 2
sin( ) sin( ) sin( ) sin( ) sin( ) 2
sin( )
2
T T
QB
γ
γ
γ γ γ γ
γ γ γ γ γγ
= =
 
 
         
       
         
 
 
we have
sin(2 ) sin(2 )
2 2
cos( ) cos( ) cos( )
2 2
sin(2 ) sin(2 ) sin( )
2 2
sin( ) sin( )
2 2
Q y Q Q x
B B A
γ β
γ β α
γ β α
γ β
= =
 
   
   
 
 
 
In a formal way we have
1 1
( ) ( )
T T T T
Q Q B B B B A A A A
BA
− −
=
Now we know that for any wave in optics we have the propagation rule or Eikonal ( field ) whose
propagation ray is always orthogonal to the tangent of the wave form. In conclusion when the form
of the wave changes the ray changes, but is always perpendicular to the tangent and so the movement
is a sequence of projection operator.
36
Figure 23 Propagation of the wave ray in geometric optics by projections into different tangent
planes in Y and Z.
In a graphic way we have
Figure 24 Wave front in the refraction as a chain of projection from x to A and from y to B = Z A
where Z is the operator by which we transform the fuzzy reference A into the fuzzy reference B.
In the physical refraction we have a chain of projections in two dimensional space. Now with the
morphogenetic projection in n dimensional space, we can simulate refraction in n dimensional
space. Refraction is a transformation of basis fuzzy set A into basis fuzzy set B = Z A.
Example 12
Given the basis A with five fuzzy sets and the basis B with again five fuzzy sets as we show as
follows
Y
X
Z
z
x
y
O
Wave front
A
Wave front
B= Z A
37
A
0.1
0.2
0.6
1
0.6
0.2
0.1
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
1
0.6
0.2
0
0
0
0
0
0
0.1
0.2
0.6
1
0.6
0.2
0.1
:= B
0.1
0.2
0.6
0.6
0.6
0.2
0.1
0
0
0
0
0
0
0.2
0.6
0.6
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
0.6
0.6
0.2
0
0
0
0
0
0
0
0.2
0.6
0.6
0.6
0.2
0
0
0
0
0
0
0.1
0.2
0.6
0.6
0.6
0.2
0.1
:=
We have the refraction operator
Refraction =
1 1
( ) ( )
T T T T
Q Q B B B B A A A A
BA
− −
=
And we have the refraction result for fuzzy sets where the input is y p , the first orthogonal projection
is QAp and the final refraction result is Refraction
0 5 10
0
2
4
y
p
p
0 5 10
0
2
4
QA
p
p
0 5 10
0
2
4
Refraction
p
p
A B
y
Figure 25 Fuzzy inference process by refraction in 11 dimensions
10. Conclusion
In this paper we present the minimum action reasoning by which we can move in the
multidimensional space with projections , reflections , rotations and refractions. For each operation
we build a special multidimensional operator and a special control by models. Because the models
are surfaces we can build a set of surfaces by which we can project , reflect or orientate rays to
control the movement from the initial point to the final point. The idea is to simulate the best
movement or action to join two points with a path controlled by surfaces with the minimum distance
( geodesic ).The multidimensional space can be of any type to cover different applications as linear
and non linear regression , fuzzy transformations, inferential fuzzy logic and many other future
applications.
38
References :
[1] Chongfu H. , Yong S. , Towards efficient fuzzy information processing , Physica – Verlag ,
A. Springer –Verlag Company 2003
[2] Diamond P. Fuzzy Least squares, Information Sciences 46(3)141-157,1988
[3] Fatmi A.,Resconi G., A new computing principle, Il Nuovo Cimento Vol.101 B,N.2, 239-
242, - Febbraio 1988
[4] Klir G. Bo Yuan , Fuzzy sets and fuzzy logic, Prentice Hall PTR New Jersey 1995
[5] Nikravesh M., Intelligent computing techniques for complex systems, in Soft Computing and
Intelligent Data Analysis in Oil Exploration, 651-672, Elsevier, 2003
[6] Resconi G. , Nikravesh M., Morphic computing, IFSA 2007 World Congress Cancun,
Mexico,June 18-21,2007
[7] Resconi G. The morphogenetic systems in risk analysis, Proceedings of the International
Conference on Risk Analysis and Crisis Response, September 25-26 , 2007.161-165 , Shangai ,
China
[8] Resconi G , A.J. van der Wal , Morphogenetic neural network encode abstract rules data ,
Information Sciences 142. 249-273, 2002
[9] Resconi, G., Nikravesh M., Morphic computing : concepts and foundation , Section in
Nikravesh M. , Zadeh L.A. , Kacprzyk J. , “Forging the new frontiers: Fuzzy Pioneers I” Springer
– Verlag in the Series Studies in Fuzziness and Soft Computing July 2007.
[10] Resconi, G., Nikravesh M., Morphic computing : quantum and field , Section in Nikravesh
M. , .Zadeh L.A , Kacprzyk J. , “Forging the new Frontiers : Fuzzy Pioneers II” Springer, in the
Series Studies in Fuzziness and Soft Computing July 2007
[11 ]Resconi G. and L. C.Jain , Intelligent Agents, Springer 2004
[12] Zadeh L.A , L.Fu , K.S. Tanaka , K.A. Shimura , Fuzzy sets and their applications to
cognitive and decision processes. Academic Press New York 1975
[13] Zadeh L.A. Toward a generalized theory of uncertainty (GTU)-an outline, Information
Sciences , Volume 172 , Issues 1-2 , 9 June 2005 , 1-40
[14] Zadeh L.A. Is there a need for fuzzy logic? , Information Sciences , Volume 178 , Issue 13 ,
1 July 2008 , 2751 – 2779
39
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