Control of the minimum action reasoning

Information Sciences xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier .com/locate / ins

Control of the minimum action reasoning

http://dx.doi.org/10.1016/j.ins.2014.02.0150020-0255/� 2014 Elsevier Inc. All rights reserved.

E-mail address: [email protected]

Please cite this article in press as: G. Resconi, Control of the minimum action reasoning, Inform. Sci. (2014), http://dx.doi.org/1j.ins.2014.02.015

Soft computing by multi dimension optic geometry

Germano ResconiFaculty of Mathematics and Physics, Catholic University, Brescia, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 February 2009Received in revised form 1 November 2013Accepted 5 February 2014Available online xxxx

Keywords:ReasoningGeodesicSoft-computingExtension of linear regressionProjection operatorGeometry of fuzzy reasoning

Given a n + 1 dimensional space S with (y, x1, x2, . . . , xn) 2 S, we use the model (hyper-plane) in S y = b1f1(x1, x2, . . . , xn) + � � � + bqfq(x1, x2, . . . , xn) to control the transformation ofthe points Y = {(y1, x1, x2, . . . , xn), (y2, x1, x2, . . . , xn), . . . , (yk, x1, x2, . . . , xn)} intoY 0 ¼ y01; x1; x2; . . . ; xn

� �; y02; x1; x2; . . . ; xn� �

; . . . ; y0k; x1; x2; . . . ; xn� ��

where the values y0 havethe minimum distance from y. The algorithm to compute the parameters (b1, b2, . . . , bq)and the values y0 is denoted minimum action reasoning. The operation is the geometricprojection of y into the hyper-plane in S. With the different models or hyper-planes wecan control many different geometric transformations as reflection, rotation, refraction.With a chain of transformations we generate the minimum path in S that joins one pointto another (geodesic). One ray in the space S is controlled by models as a special environ-ment that guides the ray to have the task with minimum distance. The minimum actionreasoning can be used to create software by models for different applications The coordi-nates of the space S can be real numbers, logic values, fuzzy sets or any other set that wecan define. Classical linear or non-linear regression is part of this minimum action reason-ing. Also classical logic, many value logic and fuzzy logic are included in the minimumaction reasoning.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The paper studies the possibility to implement minimum action reasoning in a multidimensional space S. The first step ofthe paper is to create an algorithm to project a random set of points y in S into a set of points y0 that are in the best modelwithin a family of models. The algorithm chooses among all models in the family the best model for which y and y0 have theminimum distance. The action to choose the best model is named minimum action (geodesic). Any chain of minimum actionis denoted reasoning and therefore the algorithm is denoted minimum action reasoning. More complex geometric transfor-mations are possible as projection, reflection, rotation, refraction and so on in the space S. For a family of linear models wecan use the geometric projection to compute the best linear parameters in the linear regression. We can also extend the lin-ear regression to non-linear regression or to fuzzy number transformations. The space S can be a space of logic values, asclassical logic values, many value logic and fuzzy logic.

0.1016/

http://dx.doi.org/10.1016/j.ins.2014.02.015

mailto:[email protected]


http://www.sciencedirect.com/science/journal/00200255

http://www.elsevier.com/locate/ins



2 G. Resconi / Information Sciences xxx (2014) xxx–xxx

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} wherey is a quantitative dependent variable and {x1, x2, . . . , xp} are independent variables. Regression analysis studies the depen-dence of y with respect to the variables {x1, x2, . . . , xp} when we have samples of y and samples of {x1, x2, . . . , xp}. This re-quires the choice of a suitable model and the related parameters estimation. Given the generic model:

Pleasej.ins.2

y ¼ f ðx1; . . . ; xp; bÞ þ e ð1Þ

the statistical regression aims to find the set of unknown parameters so that

~y ¼ f ðx1; x2; . . . ; xp; ebÞ ð2Þ

where ~y are the values of y in agreement with the model and with the minimum errors respect to the given samples of y. Theterm e indicates the deviation of y from the model. The most widely used regression model is the Multiple Linear RegressionModel (MLRM), as well as the Least Squares (LS) is the most widespread estimation procedure. In the MLRM the dependentvariable y would be expressed as the weighted sum of the independent variables {x1, x2, . . . , xp}, with the unknownparameters

fb1;b2; . . . ;bpg

Formally we have the hyper-plane

yn ¼ b0 þ b1xn;1 þ � � � þ bpxn;p þ en ð3Þ

where b0 is the parameter related to intercept term. In a matrix form the model is expressed as:

y ¼ Xbþ e

where

y ¼

y1

y2

. . .

yq

2666437775; b ¼

b1

b2

. . .

bp

2666437775; e ¼

e1

e2

. . .

eq

2666437775; X ¼

1 x1;1 . . . x1;p

1 x2;1 . . . x2;p

. . . . . . . . . . . .

1 xq;1 . . . xq;p

2666437775 ð4Þ

LS is based on the minimization of the sum of squared deviations:

minb

D ¼ ðy� XbÞTðy� XbÞ

where ðÞT is the matrix transposeð5Þ

The optimal solution b of the minimization problem is obtained in this way

D ¼ ðy� XbÞTðy� XbÞ ¼ yT y� yT Xb� ðXbÞT yþ ðXbÞTðXbÞ

To compute the minimum value we make the derivatives of the previous form

@D@bj¼ �yT X

@b@bj� @b

T

@bjXT yþ @b

T

@bjXT Xbþ bT XT X

@b@bj

where

b ¼

b1

. . .

bj

bjþ1

. . .

bp

2666666664

3777777775and

@b@bj¼

0. . .

10

. . .

0

2666666664

3777777775¼ v j;

@bT

@bj¼ 0 . . . 1 0 . . . 0½ � ¼ vT

j

We have

@D@bj¼ 0

for yT Xv j þ vTj XT y ¼ vT

j XT Xbþ bT XT Xv j

But because we have the following scalar property

cite this article in press as: G. Resconi, Control of the minimum action reasoning, Inform. Sci. (2014), http://dx.doi.org/10.1016/014.02.015



G. Resconi / Information Sciences xxx (2014) xxx–xxx 3

Pleasej.ins.2

P ¼ AT B ¼ ðAT BÞT¼ BT A

the previous expression can be written as follows:

vTj ðX

T yÞ ¼ vTj ðX

T yÞh iT

¼ ðXT yÞTv j ¼ yT Xv j

vTj ðX

T XbÞ ¼ vTj ðX

T XbÞh iT

¼ ðXT XbÞTv j ¼ bTj ðX

T XÞTv j ¼ bTj ðX

T XÞv j

We have

yT Xv j þ vTj XT y ¼ 2vT

j XT y

vTj XT Xbþ bT XT Xv j ¼ 2vT

j XT Xb

and

2vTj XT y ¼ 2vT

j XT Xb

whose solution is

XT y ¼ XT Xb

b ¼ ðXT XÞ�1XT y

ð6Þ

For the previous solution for the optimal condition we obtain

y ¼ XðXT XÞ�1XT yþ e ¼ Qyþ e ð7Þ

We remark that the operator Q = X(XTX)�1XT is a projection operator for which

Q 2 ¼ XðXT XÞ�1XT XðXT XÞ�1

XT ¼ XðXT XÞ�1XT ¼ Q

Geometric image of the projection operator (Fig. 1)

Example 1.

X ¼1 11 21 3

264375; y ¼

122

264375

Parameters

b ¼ ðXT XÞ�1XT y ¼

2312

" #
Projection vector
Qy ¼ XðXT XÞ�1XT y ¼

1 11 21 3

264375 3

212

" #¼

32þ 1

2 ð1Þ32þ 1

2 ð2Þ32þ 1

2 ð3Þ

264375 ¼ 2

52

3

264375

The values in the projection vector are samples of the best fit of this linear form

(1- Q)y

Qy = z

x1

x2

s 1

s 2

s 3 y

Fig. 1. Projection of the vector y into the plane of two dimensions X = (x1,x2).




Fig. 2. Best fit linear form with original point y.


Pleasej.ins.2

y ¼ 12þ 2

3x

Graph in Fig. 2 with points in the vector y with the previous linear model.

The projection of y into the plane X is the minimum path for which the three values in y (cluster of points) move to thefinal point on the straight line. Now for

z ¼ XðXT XÞ�1XT y ¼ Qy

and

p ¼ ðI � QÞy

We have that z is orthogonal to y in fact

zT p ¼ ðQYÞT ½ðI � QÞY� ¼ YT Q TðI � QÞY

but

QT ¼ ðXðXT XÞ�1XTÞ

T¼ XðXT XÞ�1

XT ¼ Q

So

QTðI � QÞ ¼ QðI � QÞ ¼ Q � Q 2 ¼ Q � Q ¼ 0

Now the projection is a segment line from y to its projection Qy which is on a straight line (minimum distance). The move-ment from y to Qy is the minimum action whose points are

yk ¼ zþ pk; where k P 1

when

k ¼ 1; yk ¼ zþ p ¼ y

when

k!1; yk ¼ QY ¼ z

Graphic image for k = 1, k = 2, k = 3, k = 4The algorithm by which we move from the initial points y to the projection Qy with the inter-media values (movement) is

denoted minimum action reasoning.We show in Fig. 5 the projection Qy, and the movement on the segment line (1 � Q) y by which we move from y to Qy

(Figs. 3 and 4).




Fig. 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 byrhombus.

Fig. 4. When we change the initial points y, the movement of the points changes as we can see in this figure.

(1- Q)y

Qy = z

X1

X2

S 1

S 2

S 3 y

Minimum action reasoning path

Fig. 5. The minimum action reasoning movement from y to the projection Q y by a straight line or ray.


Please cite this article in press as: G. Resconi, Control of the minimum action reasoning, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.015




Example 2. Given the non-linear family of models (Fig. 7)

Pleasej.ins.2

y ¼ b1f1ðxÞ þ b2f2ðxÞf1ðxÞ ¼ 1; f 2ðxÞ ¼ ð�x2 þ 4x� 2Þ

we compute the basis vectors: one for b2 = 0 and the other for b1 = 0 so we have the two dimensional plane

X ¼

x f1ðxÞ f2ðxÞx ¼ 1 1 1x ¼ 2 1 2x ¼ 3 1 1

2666437775

In a short form

X ¼1 11 21 1

264375

For the random value of y given by the vector

y ¼122

264375

we have the parameters

b ¼ ðXT XÞ�1XT y ¼

112

" #
and the projection operator
z ¼ Xb ¼ XðXT XÞ�1XT y ¼ Qy ¼

1þ 12 ð1Þ

1þ 12 ð2Þ

1þ 12 ð1Þ

264375 ¼

32

232

264375

The vector y is a linear combination of the colon vectors in X.

y ¼ b1 þ b2ð�x2 þ 4x� 2Þ ¼ 1þ 12ð�x2 þ 4x� 2Þ

y ¼yð1Þyð2Þyð3Þ

264375 ¼ 1þ 1

2 ð�12 þ 4� 2Þ1þ 1

2 ð�22 þ 4ð2Þ � 2Þ1þ 1

2 ð�32 þ 4ð3Þ � 2Þ

26643775 ¼

32

232

264375 ¼ 1:5

21:5

264375

Because

ðI � yðyT yÞ�1yTÞ

Ty ¼ ðI � ðyðyT yÞ�1

yTÞTÞy ¼ ðI � yðyT yÞ�1

yTÞy

and

ðI � yðyT yÞ�1yTÞy ¼ 0

so

ðI � yðyT yÞ�1yTÞ is orthogonal to y

Numerically we have the three vectors orthogonal to y

p ¼ ðI � yðyT yÞ�1yTÞ ¼

2534 � 6

17 � 934

� 617 � 9

17 � 617

� 934 � 6

172534

264375 ¼ p1 p2 p3½ �

The previous column vectors are orthogonal to y. In a graphic way we haveBecause

QT ¼ Q ; Q 2 ¼ Q





we have

Pleasej.ins.2

½ðI � QÞy�T Qy ¼ yTðI � Q TÞQy ¼ 0

because

ðI � QTÞQ ¼ 0

and

½ðI � QÞy�T Qy ¼ 0

So (I � Q) y is orthogonal to Qy. Numerically we have (Fig. 6)

½ðI � QÞy� ¼� 1

2

012

264375

In a graphic way we haveFor the (7) we obtain

y� XðXT XÞ�1XT y ¼ y� Qy ¼ ð1� QÞy ¼ e

and

ðQyÞTe ¼ ðQyÞTð1� QÞy ¼yT Q Tð1� QÞy ¼ yT Qð1� QÞy ¼ yTðQ � Q 2Þy ¼ 0

The error e is perpendicular to the optimal condition Qy. We remark also that

Qðy� QyÞ ¼ Qð1� QÞy ¼ Qe ¼ 0

The projection of the error is equal to zero and therefore

Qðyþ eÞ ¼ XðXT XÞ�1XTðyþ eÞ ¼ Qyþ Qe ¼ Qy ð8Þ

The projection operator Q projects the variable y + e into Qy where the error is eliminated.

3. Metric G in the parameter space

Let us prove that the minimum action reasoning can be written as follows:

y ¼ Xb

minb

P ¼ bTðXT XÞb ¼ bT Gb

E ¼ XT y

8<: ð8Þ

where P is the minimum distance (geodesic) in the space of the parameters b with a metric G. The form E = XTy is the con-strain in the minimum action reasoning that is invariant for the projection operator Q. In fact we have

XT Qy ¼ XT y ¼ E ð9Þ

Proof. To solve the minimum problem with constrains we use the Lagrange multipliers and thus

D ¼ bT Gbþ kðE� XT yÞ ¼ bT Gbþ kðE� XT XbÞ ¼ bT Gbþ kðE� GbÞ

Now we compute the derivative in this way

@D@bj¼ @b

T

@bjGbþ bT G

@b@bj� kG

@b@bj¼ 0

that can be written as follows

vTj Gbþ bT Gv j ¼ 2bT Gv j ¼ kGv j

For which k = 2bT and

DðbÞ ¼ bT Gbþ 2bTðE� GbÞ ¼ bT Gbþ 2bT E� 2bT Gb ¼ 2bT E� bT Gb





We have also that

Pleasej.ins.2

e2 ¼ ðy� XbÞTðy� XbÞ ¼ yT y� yT Xb� ðXbÞT yþ ðXbÞTðXbÞ¼ yT y� ETb� bT Eþ bT Gb ¼ yT y� 2ETbþ bT Gb ¼ yT y� DðbÞ

If D(b) assumes the minimum value also the error e2 assumes the minimum value. h

Example

D ¼ 2b1E1 þ 2b2E2 � 2b21 þ 2b2

2 þ 2b1b2

� �@D@b1¼ 2ðE1 � 2b1 � b2Þ ¼ 0

@D@b2¼ 2ðE2 � 2b2 � b1Þ ¼ 0

So we have

E1 ¼ 2b1 þ b2

E2 ¼ 2b2 þ b1

or

E1

E2

� �¼

1 01 10 1

264375

T 1 01 10 1

264375

0B@1CA b1

b2

� �¼

2b1 þ b2

2b2 þ b1

� �

or

E ¼ Gb; b ¼ G�1E

y ¼ Xb ¼ XG�1XT Y ¼ QY

In a general case we have that the minimum condition is

@D@bj¼ 2

@bT

@bjE� @bT

@bjGbþ bT G

@b@bj

!¼ 2vT

j E� 2vTj Gb ¼ 0

The solution is E = Gb for which we have

b ¼ G�1E ¼ G�1XT y ð10Þ

And for the minimum action reasoning for b we have the projection operator

y ¼ Xb ¼ X G�1XT y ¼ Qy ð11Þ

The (11) is the minimum action reasoning by projection operator. Now for the error e, for which Qe = 0, we obtain

y ¼ Xb ¼ X G�1XTðyþ eÞ ¼ Qyþ Qe ¼ Qy ð12Þ

The projection operator separates the variable y from its error e. The elimination of the error e from the original variable y inthe projection operation gives the meaning of the optimal condition for b.

We remark that the minimum action reasoning is generated by a conditional minimum with constrain (8) without the compu-tation of the variance.

Example 3. Given the column space

X ¼1 01 10 1

264375

We have (Fig. 8)

q1

q2

q3

264375 ¼ Q

y1

y2

y3

264375 ¼ 1 0

1 1

0 1

264375 b1

b2

" #¼

b1

b1 þ b2

b2

264375P ¼

b1

b2

" #T 1 0

1 1

0 1

264375

T 1 0

1 1

0 1

264375 b1

b2

" #¼

1 0

1 1

0 1

264375 b1

b2

" #0B@1CA

T 1 0

1 1

0 1

264375 b1

b2

" #

¼ ðQyÞTðQyÞ ¼ q21 þ q2

2 þ q23 ¼ b2

1 þ ðb1 þ b2Þ2 þ b2

2 ¼ 2b21 þ 2b2

2 þ 2b1b2





So in the space of the samples q the form P is a simple quadratic form and the geometry is the traditional Euclidean space. Inthe space of the parameters b the form P is a quadratic form with a cross (non-Euclidean space) term that gives the depen-dence between the two vectors

Pleasej.ins.2

x1 ¼110

264375; x2 ¼

011

264375

In a graphic way we haveWe remark that the unitary transformation U for which we have

UT U ¼ 1and

P ¼ ðUbÞTðUGbÞ ¼ bT UT UGb ¼ bT Gb

ð13Þ

is a transformation for which P is invariant. When P assumes the minimum value, the change of the parameters b and Gb by Udoes not change the minimum value P. The transformation U is the unitary transformation that gives all the possible param-eters for which we have the minimum value for P. This is similar to the least action in physics. Any change of the referencedoes not change the least action property in mechanics.

4. Minimum action reasoning in the fuzzy number space

In this chapter we suggest a new representation of the classical fuzzy inference process [1–3] by extension of the linearregression to minimum action reasoning by projection operators and to variables or coordinates whose values are fuzzynumbers.

Let ðy; x1; x2; . . . ; xnÞ 2 S where ðy; x1; x2; . . . ; xnÞ 2 S ¼ Uy � Ux1 � . . .� Uxn

Each universe Uj is a domain whose values are fuzzy numbers. The fuzzy model is

y ¼ b1f1ðx1; x2; . . . ; xnÞþ; . . .þ bqfqðx1; x2; . . . ; xnÞ

where the basis functions are functions whose independent variables and dependent variables are fuzzy numbers.

f1ðx1; x2; . . . ; xnÞ ¼ y1 2 U

f1ðx1; x2; . . . ; xnÞ ¼ y1 2 U

. . . . . . . . . . . . . . . . . .

fqðx1; x2; . . . ; xnÞ ¼ yq 2 U

where U is a collection of fuzzy numbers. In the minimum action reasoning in the real numbers we have

y ¼

y1

y2

. . .

yq

2666437775; X ¼

1 x1;1 . . . x1;p

1 x2;1 . . . x2;p

. . . . . . . . . . . .

1 xq;1 . . . xq;p

2666437775

When we substitute the ordinary numbers with the fuzzy numbers Ai,j we obtain for X the matrix

XðxÞ ¼

A1;1ðxÞ A1;2ðxÞ . . . A1;pðxÞA2;1ðxÞ A2;2ðxÞ . . . A2;pðxÞ

. . . . . . . . . . . .

Aq;1ðxÞ Aq;2ðxÞ . . . Aq;pðxÞ

2666437775

where X is the fuzzy connection matrix. We have also that for the random fuzzy number

YðyÞ ¼

B1ðyÞB2ðyÞ

. . .

BqðyÞ

2666437775

In the same way in which we have computed the parameters b in chapter 2, we can compute the functional parameters b inthe functional space of the fuzzy numbers

bðx; yÞ ¼ ðXðxÞT XðxÞÞ�1

XðxÞT YðyÞ ð14Þ





where

Pleasej.ins.2

GðxÞ ¼ ðXðxÞT XðxÞÞ ¼

PjA

2j;1ðxÞ

PjAj;1ðxÞAj;2ðxÞ . . .

PjAj;1ðxÞAj;nðxÞP

jAj;2ðxÞAj;1ðxÞP

jA2j;2ðxÞ . . .

PjAj;2ðxÞAj;nðxÞ

. . . . . . . . . . . .PjAj;nðxÞAj;1ðxÞ

PjAj;nðxÞAj;2ðxÞ . . .

PjA

2j;nðxÞ

266664377775

Given the functions (14) we can build the best output of fuzzy numbers with the expression

Yðx; yÞ ¼ XðxÞbðx; yÞ ¼ XðxÞðXðxÞT XðxÞÞ�1

XðxÞT YðyÞ

where

Eðx; yÞ ¼ XðxÞT YðyÞ

is the functional input that is connected with the coefficients (14) in the following way

bðx; yÞ ¼ ðXðxÞT XðxÞÞ�1

Eðx; yÞ ¼ GðxÞ�1Eðx; yÞ

The functions b(x,y), E(x,y) are dual functions for which we have

Pðx; yÞ ¼ bðx; yÞT GðxÞbðx; yÞ ¼ Eðx; yÞT GðxÞ�1Eðx; yÞ

And the (8) is

Yðx; yÞ ¼ XðxÞbðx; yÞ ¼ Qðx; yÞYðyÞmin

bPðx; yÞ ¼ bTðXðxÞT XðxÞÞb ¼ bT GðxÞb

Eðx; yÞ ¼ XðxÞT YðyÞ

8<:
Now given b(x,y) we have a fuzzy model by which given a set of fuzzy sets as values for the input variables (x1, x2, . . . , xn) wecan compute the associate fuzzy set for the output variable y. In fact we have
Qðy; xÞBðyÞ ¼ b1ðx; yÞA1ðxÞ þ . . . . . .þ bpðx; yÞApðxÞ

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

zðx; yÞ ¼ b1 þ b2xþ b3yþ b4xy

In the numerical way we have

X ¼

x y xy 10 0 0 10 1 0 11 0 0 11 1 1 1

26666664

37777775; y ¼

1001

2666437775

For chapter 2 we have

b ¼

b1

b2

b3

b4

2666437775 ¼ ðXT XÞ�1

XT y ¼

�1�121

2666437775

The projection operator is

z ¼ Qy ¼ Xb ¼

0 0 0 10 1 0 11 0 0 11 1 1 1

2666437775�1�121

2666437775 ¼

1001

2666437775





Pleasej.ins.2

z ¼ 1� ðxþ y� 2xyÞ ¼ 1� ðx� yÞ2

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

X ¼

0 0 0 10 1 0 11 0 0 11 1 1 1

2666437775; y ¼

1101

2666437775

And the parameters are

b ¼

b1

b2

b3

b4

2666437775 ¼ ðXT XÞ�1

XT y ¼

�1011

2666437775

So the model of inferential rule ‘‘if A then B’’ is the following

z ¼ �xþ xyþ 1 ¼ xðy� 1Þ þ 1

5.2. Many value logic and fuzzy logic models

Given the possible values

lðxÞ ¼ x1 þ x2

2;lðyÞ ¼ y1 þ y2

2

where x and y are classical logic values one and zero, we obtain the many value logic

lðxÞ 2 0;12;1

� ; lðyÞ 2 0;

12;1

�
The general composition rule is
z ¼ b0 þ b1x1 þ b2x2 þ b3y1 þ b4y2 þ b5x1y1 þ b6x1y2 þ b7x2y1 þ b8x2y2

We can simplify and generalise the previous model in this way

z ¼ b1 þb2

n

Xj

xj þb3

n

Xj

yj þb4

n

Xj

xjyj

or

z ¼ b1 þ b2hxi þ b3hyi þ b4hxyi

where hi is the average value

Example 4. For the AND operation in the classical logic we have

b1 ¼ 0;b2 ¼ 0;b3 ¼ 0;b4 ¼12

In many value model we have

z ¼ x1y1 þ x2y2

2

�
So we have the composition rule
lðp ^ qÞ ¼

z ¼ x1y1þx2y22 0 1

212 1

0 0 0 0 012 0 1

212

12

12 0 0 1

212

1 0 12

12 1

26666664

37777775
that can be separate in two AND operations



y = (1.5,2,1.5) p2

p1

p3

Fig. 6. The tree vectors (I � y(yTy)�1yT) orthogonal to the vector y.

Fig. 7.points


Pleasej.ins.2

x ^ y 0 12 1

0 0 0 012 0 1

212

1 0 12 1

2666437775 ¼ q ¼minðx; yÞ;

x ^ y 0 12 1

0 0 0 012 0 0 1

2

1 0 12 1

2666437775 ¼ q 6minðx; yÞ

6. Minimal action reasoning for fuzzy sets

Given the triangular basis of fuzzy set

AjðxÞ ¼ ljðxiÞ ¼1� jxi�uj j

Du

�jxi � ujj 6 Du

0 jxi � ujj > Du

(

in a graphic way we haveWe have the connection matrix

XðxÞ ¼

A1ðxÞA2ðxÞ

. . .

A5ðxÞ

2666437775

We show in the three white triangles the input data y; the black points are Qy, the white squares are the points (I � Q) y orthogonal to the blackQy.




x 1

x2

q 1

q 2

q3

Fig. 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 thesample space.

Fig. 9. Set of triangular fuzzy set.

0 5 100

2

43

0

Yp

100 p

Fig. 10. Trapezoid fuzzy set.

0 5 100

2

43.232

0.096

QAp

100 p

Fig. 11. Projection of the trapezoid fuzzy set into the subspace of the fuzzy sets A.


Please cite this article in press as: G. Resconi, Control of the minimum action reasoning, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.015



y

A

B

Y = B ( AT B )-1 AT y

K = A ( AT A )-1 AT y

Fig. 12. Oblique projection of y into B through A.

Fig. 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 rhombuspoints on the straight line with the minimum error. The square are the oblique projection of the samples y1.

Fig. 14. Set of fuzzy sets A.


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 foreach column (Figs. 9–11, 14 and 15)

Pleasej.ins.2




Fig. 15. Fuzzy sets B.


Pleasej.ins.2

A :¼

0:1 0 0 0 00:2 0 0 0 00:6 0:2 0 0 01 0:6 0:2 0 0

0:6 1 0:6 0:2 0:10:2 0:6 1 0:6 0:20:1 0:2 0:6 1 0:60 0 0:2 0:6 10 0 0 0:2 0:60 0 0 0 0:20 0 0 0 0:1

0BBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCA
Now given the trapezoid fuzzy set Y(x) whose values are Yp in Fig. 15 in a numerical way we have
Y ¼

01233332100

26666666666666666666664

37777777777777777777775
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 fivedimension plane embedded in the 11 sample space. So we have
QY ¼ XðxÞðXðxÞT XðxÞÞ�1

XðxÞT YðyÞ ¼

0:3030:6051:7853:2322:9112:9733:0442:0390:8360:1920:096

26666666666666666666664

37777777777777777777775

; bðx; yÞ ¼

3:026�0:1511:4821:3070:958

26666664

37777775





In a graphic way QY, whose numerical values are QAp in Fig. 16 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 Y0 that is the linear composition of the original basis fuzzy sets Theparameters b show the weight order of the fuzzy sets A given by the trapezoid input, for which we have

y p

Pleasej.ins.2

orderðAÞ ¼

A1

A3

A4

A5

A2

26666664

37777775 ¼3:0191:4821:3070:958�0151

26666664

37777775

7. Minimal action reasoning by oblique projection

In a graphic way we have the oblique projection (Figs. 12 and 13).Given the projection operator

Qy ¼ AðAT AÞ�1

AT 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 theform of the projection operator in this way

Qy ¼ AðAT AÞ�1

AT y ¼ AðAT AÞ�1

AT Py ¼ AðAT AÞ�1

AT ½BðAT BÞ�1

AT �y ¼ AðAT AÞ�1

AT y

so

P ¼ BðAT BÞ�1

AT

And P is the oblique projection.

Example 5. Given the plane in three dimensions (colon space)

0 5 100

2

4

p

0 5 100

2

4

QA p

p

0 5 102

0

2

4

Qy p

p

BAy

Fig. 16. Projection from y to B by A by oblique projection operator.

O

y

Ref y

K

A

Q y

(I- Q )y

-(I- Q )y

Fig. 17. Reflection of y in Ref y.




Fig. 18. The rhombus points are the original points, the dot points are the projection points and the squares are the points generated by the reflectionoperator.

x

y

Ref y C

A

Fig. 19. Reflection from x to y by C in A.

x

y

Ref y P x = C

Column space A constrain

E

Oblique projections

F

Fig. 20. Oblique projection and reflection.


Pleasej.ins.2

A ¼1 11 21 3

264375

The sample for y is





Pleasej.ins.2

y1 ¼122

264375

The orthogonal projection is

y ¼ AðAT AÞ�1

AT y1 ¼1:1671:6672:167

264375

Now for B = Z A we have the oblique projection

y2 ¼ BðAT BÞ�1

AT y1 ¼ ZAðAT ZAÞ�1

AT y1

So the orthogonal projection of z on the plane X is equal to the projection of y1 on the same plane X. For

Z ¼1 0 00 3 00 0 1

264375

we have

y2 ¼0:7142:5711:714

264375

In Fig. 18 we see the transformation from y1 into YR = Qy1 and also into P y = y2

Example 6. We show the fuzzy sets A and the fuzzy sets B in this way

A :¼

0:1 0 0 0 00:2 0 0 0 00:6 0:2 0 0 01 0:6 0:2 0 0

0:6 1 0:6 0:2 0:10:2 0:6 1 0:6 0:20:1 0:2 0:6 1 0:60 0 0:2 0:6 10 0 0 0:2 0:60 0 0 0 0:20 0 0 0 0:1



B :¼

0:1 0 0 0 00:2 0 0 0 00:6 0:2 0 0 00:6 0:6 0:2 0 00:6 0:6 0:6 0:2 0:10:2 0:6 0:6 0:6 0:20:1 0:2 0:6 0:6 0:60 0 0:2 0:6 0:60 0 0 0:2 0:60 0 0 0 0:20 0 0 0 0:1


In a graphic way the basis fuzzy sets A are the basis of the fuzzy sets B = ZA is
By oblique projection from the trapezoid distribution yp we have QAp by the orthogonal projection and after by obliqueprojection Q (see Fig. 17) we obtain Qyp as we can see in Fig. 21.

8. Metric G for mixed spaces A and B in minimal action reasoning

For the oblique operator where B = ZA we have the minimal condition

y ¼ Bb ¼ ZðAbÞmin

bP ¼ bTðAT BÞb ¼ bT Gb

E ¼ AT y

8<: ð15Þ

where the metric of the parameter space is G = ATB and the transformation from y to y0 is the oblique projection

Proof. To prove (15) we can repeat the same computation in (8) by Lagrange multipliers and obtain




Fig. 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 Fig. 20.


Pleasej.ins.2

D ¼¼ bT Gbþ kðE� AT BbÞ ¼ bT Gbþ kðE� GbÞ

So

@D@bj¼ 0 for k ¼ 2b

D ¼ bT Gbþ 2bTðE� GbÞ@D@bj¼ 0 for E ¼ Gb

We remark that for the constraint E = AT y, we have the same property of the classical linear regression with

E ¼ AT y ¼ ðAT BÞðAT BÞ�1

AT y ¼ ATðBðAT BÞ�1

AT yÞ ¼ AT Qy

In conclusion we have the new type of projection operator to compute the parameters of the model

y ¼ Bb ¼ B G�1AT y ¼ Qy ð16Þ

where Q is a projection operator. In fact

Q ¼ BG�1AT ¼ BðAT BÞ�1

AT

Q 2 ¼ BðAT BÞ�1

AT BðAT BÞ�1

AT ¼ BðAT BÞ�1

AT�

ð17Þ

9. Optical geometry of fuzzy reasoning

9.1. Reflection by projection operator (see Figs. 19–21)

Now we know that the reflection operator is represented by the graph.As we can see in Fig. 22 we have (see Figs. 17 and 18)

y ¼ Qyþ ðI � QÞyRef y ¼ Qy� ðI � QÞy ¼ Qyþ Qy� y ¼ ð2Q � IÞy

The reflection point of y is function of the projection operator Q.We remark that

Ref ðRef ðyÞ ¼ ð2Q � IÞð2Q � 1Þy ¼ ð4Q � 2Q � 2Q þ IÞy ¼ y

and

QRef ðyÞ ¼ Qð2Q � IÞy ¼ ð2Q � QÞy ¼ Qy




Fig. 22. Rotation by reflection.


Example 7. Given the plane in three dimensions (colon space)

Pleasej.ins.2

A ¼1 11 21 3

264375

given the sample

y1 ¼122

264375

for the reflection operator we have

Ref y1 ¼ ð2Q � IÞy1 ¼ ð2AðAT AÞ�1

AT � IÞy1 ¼

434373

264375

Example 8. For the two dimensional space of samples we have y ¼ cosðaÞsinðaÞ

� �; A ¼ cosðbÞ

sinðbÞ

� �
QðbÞ ¼ AðbÞðAðbÞT AðbÞÞ
�1AðbÞT

¼cosðbÞsinðbÞ

� �cosðbÞsinðbÞ

� �T cosðbÞsinðbÞ

� � !�1cosðbÞsinðbÞ

� �T

¼cos ðbÞ2 sinð2bÞ

2sinð2bÞ

2 sin ðbÞ2

" #

And the reflection operator is

Ref ðbÞ ¼ 2QðbÞ � I ¼cosð2bÞ sinð2bÞsinð2bÞ � cosð2bÞ

� �
Given y, we have the reflection
Ref ðbÞyðaÞ ¼ ð2QðbÞ � IÞyðaÞ ¼cosð2bÞ sinð2bÞsinð2bÞ � cosð2bÞ

� �cosðaÞsinðaÞ

� �¼

cosðaÞ cosð2bÞ þ sinðaÞ sinð2bÞcosðaÞ sinð2bÞ � sinðaÞ cosð2bÞ

� �
Numerical example 9
a ¼ 60;b ¼ 45

We have

Ref ðbÞyðaÞ ¼cosð2ð45ÞÞ sinð2ð45ÞÞsinð2ð45ÞÞ � cosð2ð45ÞÞ

� �cosð60Þsinð60Þ

� �¼

ffiffi3p

212

" #¼

cosð30Þsinð30Þ

� �

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).





Now given the points

Pleasej.ins.2

x ¼cosðaÞsinðaÞ

� �; y ¼

q cosðbÞq sinðbÞ

� �; A ¼

cosðcÞsinðcÞ

� �
we compute the reflection point C
Ref y ¼ ð2Q � IÞy ¼cosð2cÞ sinð2cÞsinð2cÞ � cosð2cÞ

� � q cosðbÞq sinðbÞ

� �¼ q

cosð2cÞ cosðbÞ þ sinð2cÞ sinðbÞsinð2cÞ cosðbÞ � cosð2cÞ sinðbÞ

� �
With the graph
Now we compute the vector F whose origin is in x and the end in Px

F ¼ x� Ref y ¼ x� ð2Q � IÞy ¼cosðaÞsinðaÞ

� �� q

cosð2dÞ cosðbÞ þ sinð2dÞ sinðbÞsinð2dÞ cosðbÞ � cosð2dÞ sinðbÞ

� �
The vector orthogonal to the vector F is
E ¼ I � FðFT FÞ�1FT

In fact we have

EF ¼ ðI � FðFT FÞ�1FTÞF ¼ 0

With the expression of F we have

E ¼2R sin aþb

2 �dð Þ�2R sin a�b2 �dð Þþsin ðaÞ2þR2 sin ðb�2dÞ2

R2þ4 sin aþb2 �dð Þ�2Rþ1

½cosðaÞ�R cosðb�2dÞ�½ðsinðaÞþR sinðb�2dÞ�R2þ2 cosðaþb�2dÞRþ1

264375

The point C is given by the oblique projection whose operator was computed in the previous chapter

C ¼ AðET AÞ�1ET

or

C ¼R cosðcÞ½2 cosð2aþb�2cÞ�2 cosðb�2cÞþ2R cosðaþ2b�4cÞ�2R cosðaÞ�

DetR sinðcÞ½2 cosð2aþb�2cÞ�2 cosðb�2cÞþ2R cosðaþ2b�4cÞ�2R cosðaÞ�

Det

" #

where

Det ¼ 2½cosð2a� cÞ � cosðcÞ þ R2 cosð2b� 3cÞ þ R cosðaþ b� cÞ � 2R cosða� bþ cÞ þ R cosðaþ b� 3cÞ � R2 cosðcÞ

Example 10. Given

x ¼221

264375; y ¼ 1

22

264375

and

A ¼1 11 21 3

264375

Now we want to compute the minimum action from x to y by C. Before we use the reflection operator to obtain ref (y) seeFig. 25.

Ref y ¼ ð2Q � IÞy ¼ ð2AðAT AÞ�1

AT � IÞy ¼

434373

264375

after we compute the segment that joins x with the ref (y).




Y

X

Z

Fig. 23. Propagation of the wave ray in geometric optics by projections into different tangent planes in Y and Z.

z

x

y

O

Wave front A

Wave front B= Z A

Fig. 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 fuzzyreference A into the fuzzy reference B.

0 5 100

2

4

y p

p0 5 10

0

2

4

QA p

p0 5 10

0

2

4

Refractionp

p

BAy

Fig. 25. Fuzzy inference process by refraction in 11 dimensions.


Pleasej.ins.2

x� Ref y ¼

2323

� 43

264375

after we compute the vectors E

E ¼ I � ðx� Ref yÞððx� Ref yÞTðx� Ref yÞÞ�1ðx� Ref yÞT ¼

56 � 1

613

� 16

56

13

13

13

13

264375

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





Pleasej.ins.2

E ¼

56 � 1

6

� 16

56

13

13

264375

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

C ¼ AðET AÞ�1Ex ¼

535353

264375

where C is the wanted result. We show in 26 the vectors x, C, y.

9.3. Rotation by reflection and projection

We know that the rotation is the composition of two reflections as follows (see Fig. 22)

Ref ðbÞRef ðaÞ ¼ Rotð2ðb� aÞÞ

When we establish this angle of rotation b � c we have

b� c ¼ 2ðb� aÞÞ

So

b�c2 ¼ b� a

anda ¼ b� b�c

2 ¼bþc

2

We can decompose the rotation into two reflections. In fact we have

Ref ðaÞ ¼ Refðbþ cÞ

2

�¼

cosðbþ cÞ sinðbþ cÞsinðbþ cÞ � cosðbþ cÞ

� �
And now we have
Ref ðbÞRefbþ c

2

�¼

cosð2bÞ sinð2bÞsinð2bÞ � cosð2bÞ

� �cosðbþ cÞ sinðbþ cÞsinðbþ cÞ � cosðbþ cÞ

� �¼

cosðb� cÞ � sinðb� cÞsinðb� cÞ cosðb� cÞ

� �¼ Rotðb� cÞ

In a graphic way we haveRemarkEach rotation can be written as a composition of projection operatorsIn fact we have

Ref ðbÞRef bþc2

� �¼ ð2QðbÞ � IÞ 2Q bþc

2

� �� I

� �¼ I þ 4QðbÞQ bþc

2

� �� 2QðbÞ � 2Q bþc

2

� �¼

cosð2bÞ sinð2bÞsinð2bÞ � cosð2bÞ

� �cosðbþ cÞ sinðbþ cÞsinðbþ cÞ � cosðbþ cÞ

� �¼

cosðb� cÞ � sinðb� cÞsinðb� cÞ cosðb� cÞ

� �¼ Rotðb� cÞ

With reflection and rotation we have all possible cases

Ref ðhÞ Ref ð/Þ ¼ Rotð2ðh� /ÞÞ;RotðhÞ Rotð/Þ ¼ Rotðh� /Þ;RotðhÞ Ref ð/Þ ¼ Ref ð/þ h=2Þ;Ref ð/Þ RotðhÞ ¼ Ref ð/þ h=2Þ:

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

Rotðh1; h2; . . . hnÞ ¼ Rotðh1ÞRotðh2Þ; . . . RotðhnÞ

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

Pleasej.ins.2

Rotðh1Þ ¼

cosðh1Þ � sinðh1Þ 0

sinðh1Þ cosðh1Þ 0

0 0 1

266664377775

Rotðh2Þ ¼

cosðh2Þ 0 � sinðh1Þ

0 1 0

sinðh1Þ 0 cosðh1Þ

266664377775

Rotðh3Þ ¼

1 0 0

0 cosðh3Þ � sinðh3Þ

0 sinðh3Þ cosðh3Þ

266664377775

Each rotation can be decomposed into two reflections and four projection operators. Each three dimensions complex rotationcan be decomposed into 6 reflections and 6 projections.

9.4. Refraction as composition of projection operators (see Figs. 23–25)

With

x ¼cosðaÞsinðaÞ

� �; A ¼

cosðbÞsinðbÞ

� �; B ¼

cosðcÞsinðcÞ

� �

the first projection from x to y into the space A is

Q A ¼cosðbÞsinðbÞ

� �cosðbÞsinðbÞ

� �T cosðbÞsinðbÞ

� � !�1cosðbÞsinðbÞ

� �T

¼cos ðbÞ2 sinð2bÞ

2sinð2bÞ

2 sin ðbÞ2

" #

So we have

y ¼ QAx ¼cos ðbÞ2 sinð2bÞ

2sinð2bÞ

2 sin ðbÞ2

" #cosðaÞsinðaÞ

� �

And with

Q B ¼cosðcÞsinðcÞ

� �cosðcÞsinðcÞ

� �T cosðcÞsinðcÞ

� � !�1cosðcÞsinðcÞ

� �T

¼cos ðcÞ2 sin ðcÞ2

2

sin ðcÞ22 sin ðcÞ2

24 35
we have
Q By ¼ Q BQ Ax ¼cos ðcÞ2 sinð2cÞ

2sinð2cÞ

2 sin ðcÞ2

" #cos ðbÞ2 sinð2bÞ

2sinð2bÞ

2 sin ðbÞ2

" #cosðaÞsinðaÞ

� �

In a formal way we have

QBQ A ¼ BðBT BÞ�1BT AðAT AÞ

�1AT

Now we know that for any wave in optics we have the propagation rule or Eikonal (field) whose propagation ray is alwaysorthogonal to the tangent of the wave form. In conclusion when the form of the wave changes the ray changes, but is alwaysperpendicular to the tangent and so the movement is a sequence of projection operator.

In a graphic way we have.





In the physical refraction we have a chain of projections in two dimensional space. Now with the morphogenetic projec-tion in n dimensional space, we can simulate refraction in n dimensional space. Refraction is a transformation of basis fuzzyset A into basis fuzzy set B = ZA.

Example 12. Given the basis A with five fuzzy sets and the basis B with again five fuzzy sets as we show as follows

Pleasej.ins.2

A :¼

0:1 0 0 0 00:2 0 0 0 00:6 0:2 0 0 01 0:6 0:2 0 0

0:6 1 0:6 0:2 0:10:2 0:6 1 0:6 0:20:1 0:2 0:6 1 0:60 0 0:2 0:6 10 0 0 0:2 0:60 0 0 0 0:20 0 0 0 0:1



B :¼

0:1 0 0 0 00:2 0 0 0 00:6 0:2 0 0 00:6 0:6 0:2 0 00:6 0:6 0:6 0:2 0:10:2 0:6 0:6 0:6 0:20:1 0:2 0:6 0:6 0:60 0 0:2 0:6 0:60 0 0 0:2 0:60 0 0 0 0:20 0 0 0 0:1


We have the refraction operator.
Refraction = QBQA = B(BTB)�1BTA(ATA)�1AT

And we have the refraction result for fuzzy sets where the input is yp, the first orthogonal projection is QAp and the finalrefraction result is Refractionp.

10. Conclusion

In this paper we present the minimum action reasoning by which we can move in the multidimensional space with pro-jections, reflections, rotations and refractions. For each operation we build a special multidimensional operator and a specialcontrol by models. Because the models are surfaces we can build a set of surfaces by which we can project, reflect or orien-tate rays to control the movement from the initial point to the final point. The idea is to simulate the best movement or ac-tion to join two points with a path controlled by surfaces with the minimum distance (geodesic). The multidimensionalspace can be of any type to cover different applications as linear and non-linear regression, fuzzy transformations, inferentialfuzzy logic and many other future applications.

References

[1] H. Chongfu, S. Yong, Towards Efficient Fuzzy Information Processing, Physica – Verlag, A. Springer –Verlag Company, 2003.[2] P. Diamond, Fuzzy least squares, Inf. Sci. 46 (3) (1988) 141–157.[3] A. Fatmi, G. Resconi, A new computing principle, Il Nuovo Cimento 101 (B,N.2) (1988) 239–242. Febbraio.


http://refhub.elsevier.com/S0020-0255(14)00119-4/h0005






Date post:	30-Dec-2016
Category:	Documents
Upload:	germano
View:	213 times
Download:	0 times

Control of the minimum action reasoning

Documents