Научная статья на тему 'Синтез обобщеных нейронных элементов с помощью матриц толерантности'

Синтез обобщеных нейронных элементов с помощью матриц толерантности Текст научной статьи по специальности «Математика»

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Ключевые слова
МАТРИЦА ТОЛЕРАНТНОСТИ / MATRIX OF TOLERANCE / ЯДРО БУЛЕВОЙ ФУНКЦИИ / ХАРАКТЕР ГРУППЫ / GROUP CHARACTER / СПЕКТР БУЛЕВОЙ ФУНКЦИИ / SPECTRUM OF THE BOOLEAN FUNCTION / NUCLEUS OF THE BOOLEAN FUNCTION

Аннотация научной статьи по математике, автор научной работы — Geche F., Mulesa O., Buchok V.

На основании матриц толерантности и ядер булевых функций установлен критерий реализуемости функций алгебры логики на одном обобщенном нейронном элементе относительно произвольной системы характеров. Получен ряд необходимых и достаточных условий реализуемости булевых функций на одном обобщенном нейронном элементе и на основании достаточных условий разработан эффективный алгоритм синтеза целочисленных обобщенных нейронных элементов с большим числом входов

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Synthesis of generalized neural elements by means of the tolerance matrices

Application of neuromorphic structures in various spheres of human activity on the basis of generalized neural elements will become possible if effective methods for verifying realizability of the logic algebra functions by one neuron element with a generalized threshold activation function and synthesis of such elements with a large number of edntries are developed. A notion of nucleus of Boolean functions in relation to a given system of characters was introduced and algebraic structure of nuclei and reduced nuclei of Boolean functions was investigated. Relation between the nuclei of the logic algebra functions which are realized by one generalized neural element and matrices of tolerance was established. It was shown that the Boolean function is realized by one generalized neuron element if and only if the nucleus of this function admits representation by the matrices of tolerance. If there is no nucleus relative to a specified system of characters for a Boolean function, then such a function is not realized by one generalized neural element in relation to a specified system of characters. On the basis of the properties of the matrices of tolerance, a number of necessary and sufficient conditions for realization of the logic algebra functions by one generalized neural element were obtained. Based on the sufficient conditions, an algorithm for synthesis of integer-valued generalized neural elements with a large number of entries was constructed. In the synthesis of integer-valued generic neural elements for realization of the logic algebra functions, a block representation of the Boolean function nucleus was used and based on the properties of the matrices of tolerance, coordinates of the integer vector of the structure of the generalized neural element were sequentially found

Текст научной работы на тему «Синтез обобщеных нейронных элементов с помощью матриц толерантности»

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На основi властивостей матриць толерантно-стi i ядер булевих функцш встановлено критерш реалiзованостi функцш алгебри логжи одним уза-гальненим нейронним елементом видносно довшь-ног системи характевiв. Отримано ряд необхдних та достаттх умов реалiзованостi булевих функцш одним узагальненим нейронним елементом i на основi достаттх умов розроблено ефективний алгоритм синтезу щлочислових узагальнених ней-ронних елементiв з великим числом входiв

Ключовi слова: матриця толерантностi, ядро булевог функци, характер групи,спектр булевог функцш

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На основании матриц толерантности и ядер булевых функций установлен критерий реализуемости функций алгебры логики на одном обобщенном нейронном элементе относительно произвольной системы характеров. Получен ряд необходимых и достаточных условий реализуемости булевых функций на одном обобщенном нейронном элементе, и на основании достаточных условий разработан эффективный алгоритм синтеза целочисленных обобщенных нейронных элементов с большим числом входов

Ключевые слова: матрица толерантности, ядро булевой функции, характер группы, спектр

булевой функции -□ □-

UDC 681.5:519.7

|DOI: 10.15587/1729-4061.2017.108404|

SYNTHESIS OF GENERALIZED NEURAL ELEMENTS BY MEANS OF THE TOLERANCE MATRICES

F. Geche

Doctor of Technical Sciences, Professor, Head of Department* E-mail: fedir.geche@uzhnu.edu.ua O. M ulesa PhD, Associate Professor* E-mail: oksana.mulesa@uzhnu.edu.ua V. Buchok Postgraduate student* E-mail: vityabuchok@gmail.com *Department of cybernetics and applied mathematics Uzhgorod National University Narodna sq., 3, Uzhhorod, Ukraine, 88000

1. Introduction

Intensification of theoretical development and practical applications has been observed recently in the field of information technology and neurocomputers. This is due to the increased interest in information systems and neurolike structures that have found wide application in encription, protection of information, image recognition, forecasting and other fields of human activities.

Solving complex applied problems using neuromorphic structures will become more effective when generalized neural elements (GNE) (which by their functional capabilities exceed classical neural elements) with threshold functions of activation will be used as basic elements. Therefore, information processing in the neurobase will be more effective on condition that generalized neural elements are used. To this end, it is necessary to devise practically suitable methods for the synthesis of neural elements with generalized threshold functions of activation and synthesis of logic circuits from them.

Relevance and practical value of development of new methods for the synthesis of generalized neural elements are evidenced by an increasing volume of investments in software and hardware for artificial intelligence. It should be mentioned that an extremely important requirement to the new methods of synthesis of generalized neural elements is that these methods should be practically suitable for synthesizing GNE with a large number of inputs. This is explained by the fact that the volume of information and the degree of complexity of the tasks that are solved in the neurobase are

constantly growing. That is why the studies giving results applicable in synthesizing generalized neural elements with a large number of inputs appear to be topical. The application of generalized neural elements can reduce the number of artificial neurons in the neural networks employed for the tasks on recognition, compression and encoding of discrete signals and images.

2. Literature review and problem statement

Currently, neurolike structures are increasingly used to solve varied applied problems. An indication of this is the increase in the number of scientific publications and new methods of training (synthesis) of neural networks used in various spheres of human activities. Development of new methods for data processing in the neurobase is a relevant and practically important task. For example, [1] introduces the concept of the operational base of neural networks and shows its application in the development of effective data processing methods. Possibility of using artificial real-time neural networks in the problems on digital signal processing is considered in article [2], while paper [3] investigates the feasibility of employment of neuromorphic structures for solving prediction problems in the field of intelligent data analysis.

The field of practical applications of neural network models is vast. These models are effectively used to improve resolution of images based on artificial neural networks [4], for segmentation [5], classification and pattern

recognition [6, 7]. On the basis of neural networks, intelligent blocks of various systems for controlling chemical processes [8] and for the classification of diseases [9] are developed. These models are successfully used in diagnostics [10], economic [11] and biological process [12] forecasting and morbidity prediction for the diseases under study [13]. As the studies show, neural network methods are widely applied for the compression of discrete signals and images [14-16] and in the banking sector for credit risk assessment [17].

It should be mentioned that various iterative methods and methods of approximation of various orders form the basis for construction of neural networks for the above spheres of human activities. These methods solve the tasks of training one neural element with varied functions of activation and training neural networks consisting of these elements with a certain accuracy. However, there are problems for which approximate solutions are unacceptable, for example, the problem of feasibility of Boolean and multivalued logic functions by a single neural element with a threshold function of activation or a generalized neural element relative to a specified system of characters and in the synthesis of combinational circuits from the mentioned neural elements. These combinational circuits can be successfully used in the construction of functional blocks of logical devices for controlling technological processes, compression of discrete signals, recognition of discrete images and so on. Disadvantages of approximation methods and iterative methods of training neural elements and neural networks for solving problems on the implementation of Boolean and multiple-valued logic functions by a single neural element (neural network) are as follows:

- instead of an exact solution, one obtains an approximate solution of the problem (for example, a discrete function is implemented by one generalized neural element and the approximation method and iterative methods show its unreliazability relative to the prescribed accuracy (here a problem appears about choosing exactness, the order of approximation and convergence of the process of training the generalized neural element relative to the prescribed accuracy));

- ability of applying methods of approximation and iterative methods of training artificial neurons with just a small number of inputs (up to 50) whereas biological neurons can have thousands of entries.

Given this, a development of methods for checking re-alizability of Boolean functions by one generalized neural element relative to an arbitrary system of characters should be recognized as promising. Solutions on the synthesis of corresponding generalized neural elements under certain restrictions on their nuclei can be used in a case when application of approximation and iterative methods is inexpedient or practically impossible.

3. The aim and objectives of the study

The study objective was to develop effective methods for verifying realizeability of the logic algebra functions by one generalized neural element and methods for the synthesis of generalized neural elements with integer structural vectors. On the basis of these elements, one can develop logical blocks of different devices for solving problems of practical importance in the field of compression and transmission of

discrete signals, recognition of discrete images, diagnosis of technical devices.

To achieve this goal, it was necessary to implement the following:

- to establish a criterion of realizability of Boolean functions by one generalized neural element;

- to ensure such necessary conditions for realizeability of the logic algebra functions by one neuron element with a generalized threshold activation function which would be easily verified;

- to obtain sufficient conditions for realizeability of the logic algebra functions by one generalized neural element by establishing which algorithm of the synthesis of integer generalized neural elements is constructed.

4. Mathematical model of neural elements with a generalized threshold activation function and their application when implementing Boolean functions

4. 1. The criterion for implementing Boolean functions by one generalized neural element

Let H2={-1,1} be a cyclic group of second order, Gn = H2 ®... ® H2 is the direct product of n cyclic groups H2, and x(Gn) is the group of characters [18] of the group Gn over the field of real numbers R. Use the set R\{0} to define the function:

Rsign x =

1, if x>0, -1, if x <0.

(1)

Let Z2={0,1}, ie{0, 1,..., 2n-1} and (ii,...,in) is its binary code, i. e.

i = i12n—1+i22n—2+. + in, ij e {0,1}.

Values of character x on the element

g = ((-1)a1,...,(-1)an )eGn

((a1,.,an)eZ2n - n-and the Cartesian degree Z2) are assigned as follows:

X (g) = ( — 1)a1l1+a2'2+--+an'n (2)

Consider the 2n-dimensional vector space

Vr = {M: Gn ^ R}

over the field R. Elements

X i (i = 0,1,2.....2n — 1)

of the group X(Gn) form an orthogonal basis of the space VR [18]. The Boolean function in alphabet {-1,1} specifies a single-valued transformation f: Gn ^ H2, i. e. feVR. Consequently, the arbitrary Boolean function feVR can be uniquely written in the form:

f (g) = Wg) + siXi(g) + • ■ ■ + s2„ _1X2„-1(g).

(3)

Vector sf = (s0,s1v..,s n ) is called a spectrum of the Boolean function f in the system of characters x(Gn ) (in the system of Walsh-Hadamard basis functions [19]).

Taking various characters exept the main one, construct an m-element set X=(X^,•••, X } and consider the mathematical model of the neural element with the generalized threshold function of activation relative to the chosen system of characters:

f (^(g),..., x„ (g)) = Rsign

. j=i

(4)

x=K }c * (G),

and if so, then how its structure vector can be found? Using transformation

1

x' = —(x + 1),

2 7

realize mapping {—1,1}—>{0,1} and consider the system

=4 fo +1), 4 =2 (x,2+1),..., x;m =2 (x,m+.

Let

f -4(-1) = {g eG„|f (g) = -1}

and

With the help of the system x', find:

f(-1(-1)= U {(x;(g),.,x'm(g))},

g<f-1(-1)

fx-1(1)= U {fete) —X'm (g))} .

where vector w = (ra1v..,ram;ra0) is called the vector of the structure of the generalized neural element with respect to the system of characters x and g e Gn.

If there exists a vector satisfying equation (4) for the given function f:Gn®H2 and the system X = (X^,—, X }, then it is said that function f is realized by one generalized neural element relative to x

It is obvious that the neural element in relation to the system of characters X=(X1,X2,X4,---,X n-1} coincides with the neural element with a threshold activation function (with a threshold element [20]). For an arbitrary Boolean function f: Gn ^ H2, one can always choose such a system X that the generalized neural element realizes the function f relative to x Indeed, if no limitations are imposed on the number of entries of generalized neural elements, then a system of characters x(Gn )\X0. can be chosen as x Then it follows from (3) and (4) that an arbitrary Boolean function is realized by one generalized neural element with a structure vector coinciding with the spectrum of the function in the system of Walsh-Hadamard basis functions. Further, in addition to this trivial case, in order to reduce the number of entries to the GNE, consider systems x that do not coincide with x( Gn )\X0. Obviously, the less elements in the x system, the more efficiently these elements can be used in neural networks for compressing, transmitting and recognizing discrete signals and images.

Let

f ( x1vv Xn )

be the Boolean function in alphabet (-1,1}, that is f:Gn®H2. Consider the problem whether function f (x1v.., xn) is realized by one GNE relative to the system of characters

f -4(1) = {g eG„|f (g) = 1}.

gef-1(1)

The nucleus of the Boolean function f (x1,.,xn) with respect to the system of characters X of the group Gn is defined as follows:

K(fx H x , ,

X Ifx-1(-1), if fx-1(1)> f(-1)(-1),

fx-1(1) n fx-1(-1) = 0,

where |fx-1(i)| is the number of elements of the set

fx-1(i) (i e (-1,1}).

If fX-1(1) n fX-1(-1) = 0, than nucleus K(fx) does not exist and this means that function f is not realized by one GNE relative to the system x. Let

K (fx )={a1=(a1,..., a!),..., a? = (aq,..., a qn)}.

Construct the reduced nucleus K(fx ) relative to the element

a, =(a1 ,.,a ; )eK f)

and a set of reduced nuclei T (fx ) as follows:

K ( fx ), = a,K ( fx ) =

= ((a1 ©a^..., a'n ©an )|(a1,..., an) eK (f%)},

T (fx ) = {K (fx), = a,K (fx )li q}

where © is the sum by modulus 2.

Let Z2 = {0,1} and Z2n be the n-th Cartesian power of the Z2 set. If the following conditions are fulfilled for the function f : Gn ^ h2 and the system x=(x. w x, }

1 m

fx-1(1) n fx-1(-1) = 0 and Zm * fx-1(1) U fx-1(-1),

then function f is a partially defined function in the group Gm and a concept of a extended nucleus with respect to the system of characters x is introduced for such functions as follows: let K(fx) = {a1v..,aq} be a nucleus of the Boolean function f with respect to X=(Xi ,•••,Xi } and fx-1(*) is the sets of those nests with Z2m for which function was not defined, then the following will be implied: under the extended nucleus of function f relative to the system x:

K ( fx, s) = {a1,..„ a?, b1,., bs},

where b1,.,bs are the arbitrary elements of the set fX-1(*) and q + s < 2m-1. It should be mentioned [20] that the Bool-

if

ean function f is simultaneously realized or is not simultaneously realized by one neural element in different alphabets

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{—1,1} ({0,1}).

If notation b1 = aq+1,...,bs = aq+s, is introduced, then the set of reduced nuclei of the Boolean function f with respect to the system x= {Xi ,•••,Xi } is specified as follows:

1 m

T ( U, s ) = K ( f%, s)i = = aK ( fx, s)|i = 1,2,., q + s}.

Let Em be the set of tolerance matrices [21] and

K f ) = {a1,.,a q}

is the nucleus of the Boolean function f:Gn®H2 with respect to the system of characters X= {X^,•••,Xi } of the group Gn over the field R. Use elements of the nucleus K(fX ) to construct matrix K^ (fX ) as follows: the first row of the matrix will be the vector

a«1) = (al;(1)1,•■■, a^(1)n )

with K(fX), and the second row of the matrix will be vector

different ((x;, w) is the scalar product of the vectors x;, w) and LeEm. Denote by h(L(q)) the set of m-dimensional Boolean vectors constructed from the rows of the matrix L(q). Obviously,

K ( fx ) = h(K% ( fx )) for all Çe Sq.

If the function f:Gn®H2 is realized by one GNE with respect to the system

X ÎXi , • ■ ■, Xi }

1 m

then in accordance with [22], this function is realized in alphabet Z2 as well. This means that there exists such vector

w = (<»,,...,ra )eW ,

v 1 " " m' m'

that satisfies one of the following conditions: if

K( fx, 5) = /z-1(1),

then

*Ç(2)

aÇ(2)1v,a

VX îK(/x, 5)

the last row of K^ (/x ) will be

*%(q )1

,•••, a

%(q )n)

where £,(i) is the effect of substitution of e Sq for i. Denote the first r lines of the matrix L eEm by L(r) and enter the concept of representation of the nucleus K(fX) (extended nucleus K( f%,s)) with the matrices of tolerance with Em as follows: if there is such an element ^eSq and such a matrix L e Em that K(f) = L(q) (K(f,s) = L(q)), then the nucleus K(f%) (extended nucleus K(fX,s)) admits representation by the matrices of tolerance with Em .

Theorem 1. The Boolean function f:Gn®H2 is realized by one generalized neural element with respect to the system of characters X ={X i, • • •, X i } of the group Gn over the field R, if and only if one of the conditions exists and is fulfilled:

1) the nucleus K(fX ) admits representation by the matrices of tolerance with Em and

Z2m = f—1(1) u f^1(—1);

2) the nucleus K( f ) or extended nucleus K(fX, s) admits representation by the matrices of tolerance with Em and

zm * f—1(1) u Ux-1(—1).

Proving. In the case

Z2m = fx—1(1) u fx—1(—1)

provided that K(fX ) exists, the theorem is proved analogous to Theorem 1 in [21].

Let

Z™ * fx 1(1) u fx 1(—1),

Wm be the set of all m-dimensional real vectors w such that for all different x1,x2 eZ2m, numbers (x1, w) and (x2, w) are

and

Vy ez; \ K(/x,5) (x, w) > (y, w),

in the opposite case, (K ( /x ,s) = / _1(-1))

(5)

Vx eK ( /x, 5)

and

Vy ez; \ K ( / , 5) (x, w) < (y, w).

(6)

On the basis of (5) and the properties of the matrices of tolerance [21], it can be asserted that there exists such matrix of tolerance Lw eEm, that

K ( fx, s) = h(Lw (q + s )).

In the case of (6), we have

K( fx ,s) = h(LWt(q + s)),

where w1 = —w. Thus, K( f,s) admits representation by Hoy matrices of tolerance with Em, so the necessity is proved.

Let K(f,s) admits representation by the matrices of tolerance with Em, that is, there is such matrix

L = (a, ) eEm (i = 1,2,...,2;-1; j = 1,2,..., ;),

that

K( f,s) = h(L(q + s)).

Make the matrix of tolerance L*=(asj) to correspond to the matrix of tolerance L = (as>-) in the following way: s = 2n—1 — i +1 and asj = a,j (a,j is an inverted value of ai). Define operation v for the matrices of tolerance L and L as follows:

L V L

According to the construction of the set Em [21], there exists such vector

w = (ra1,...,ra )eW ,

v 1" " m' m'

for an arbitrary matrix LeEm that

(L V L* )■ wT = cw, (7)

where

cw = (c1,c2, c3..., ct)(c1 > c2 > c3 > ... > ct;t = 2m ).

It follows immediately from equation (7) and equality

K( f,s) = h(L(q + s))

that there exists such vector w in the set Wm which satisfies inequality (5). This means that the generalized neural element relative to the system of characters x' with the weight vector w in alphabet Z2 realizes function f or f. These functions, either are simultaneously realized or simultaneously not realized by one generalized neural element. Sufficiency is proven. Consequently, the theorem is proved completely. Let

W— ={w = (ra1v..,ra )eW |0>ra1>...>ra }

ml v1" " n' m i 1 m>

and

Em = U Lw,

weW

m

where

(Lw V L*w )■ wT = cw.

Use the notion of reduced nuclei of the Boolean function f:Gn®H2 with respect to the system of characters

4. 2. Conditions necessary for the implementation of Boolean functions by one generalized neural element

Verification of the conditions necessary for realization of Boolean functions by one generalized neural element is an important step in the GNE synthesis. With the help of these conditions, functions of the logic algebra can be identified at the initial stage of synthesis of generalized neural elements which cannot be realized by one GNE in relation to a specified system of characters.

Theorem 3. If the boolean function f:Gn®H2 is realized by one generalized neural element with respect to the system of characters x = {X^, • ■ ■, X; } of the group Gn over the field R, then there is a nucleus K(fx) and the following takes place:

a = (a1.....am)eK(f )--

= (a1,...,a m )e K(fx), (8)

where ai is the inverted value a,.

Proving. As it follows from Theorem 1, there is a nucleus K(fx ) or an extended nucleus K(fx, s) that admits representation by the matrices of tolerance with Em. Consequently, there exists such a matrix of tolerance LeEm, and such an element £,e Sq or an element £,eSq+s, that one of equalities

K%(fx) = L(q) or K%(fx,s) = L(q)

is satisfied. Rows of the matrix L are the elements of a certain class of tolerance relative to

x:(a1,..., an )x(P1,.,Pn) ^3i(a, = p,).

It follows from this that the prematrix of tolerance L(q) does not concurrently contain vectors

a = (a1,., am), a = (a1,., am),

where a = (a1v..,a m ) is an arbitrary row of the matrix L(q), so the theorem is proved. The vector

a = (a1.....am )eZm

precedes the vector

X {Xi. , • " , Xi }

1m

of the Gn group and obtain the following on the basis of Theorem 1 and equality

Em = {(gL)°|L eEm, g eGm, oeS„} [21]:

Theorem 2. The Boolean function f:Gn®H2 is realized by one generalized neuron element relative to the system of characters x ={x i, • • •, X i } of the group Gn over the field R if and only if there is X( fx ) O^nd one of the following conditions is fulfilled:

1) there is at least one such element K(fx )i, in the set of reduced nuclei T(fx) that admits representation by the matrices of tolerance with E— and

m

Z2m = fx—1(1) u fx—1(—1);

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2) there is at least one such element K( fx )i in the set of reduced nuclei T(fx) or such element K(fx, s),, in a set of extended reduced nuclei T(fx, s) which admits representation by matrices of tolerance with E- and E-.

mm

b = (P1,„.,Pm)eZm (a^ b), if ai <ft (i = 1,2,...,m).

Denote by Ma the set of all such vectors with Zm, that precede the vector a.

Theorem 4. If the Boolean function f:Gn®H2 is realized by one generalized neural element with respect to the system of characters x = {X, vvX, } of the group Gn over the field R, then there is a nucleus K(fx) and one of the following conditions is fulfilled:

1) if Z2m = fX—1(1) u fX—1(—1), then there is such an element K(fx )i, in the set of reduced nuclei T(fx ) that

VaeK(fx)i ^Ma cK(fx),; (9)

2) if Z2m * fX—1(1) u fX—1(—1), then either there is such an element K( f )i, in the set of reduced nuclei T(fx) which satisfies condition (8) or it is possible to construct a set of extended reduced nuclei T(fx, s), which contains such an element K(fx, s), that

Va eK (fx, s); ^ Ma c K (fx, s), . (10)

The proving of this theorem follows directly from the rules of constructing the set of extended reduced nuclei T(fx,s), of Theorem 2 and Theorem 3 in [23].

Suppose B = (Pkr) is a rectangular q x m matrix over Z2, A c Z2m, e; is a unit vector the i-th coordinate of which is equal to 1,

n

n( A) = (i | es e A}, k(B) = Jpkr

r 1

and |A| is the number of elements of the set A.

Theorem 5. If the Boolean function f:Gn®H2 is realized by one generalized neural element with respect to the system of characters x =(x ■ ,—,X i } of the group Gn over the field R, then there is a nucleus K\fx) = {a1,.,aq} and the folowing takes place:

1) if

ra1 = -1, ra; = - 1(i = 2,3,.,m)

j=1

Z2m = fx-1(1) u fx-1(-1)

K(fx, s);, that

3) 2j < q + s < 2j+1( j s{1,2,., m - 2});

4) Vk e {1,2...,q + s} k(K(fs).) < j +1;

5) |n(K( fx, s)J> j +1.

(13)

(14)

(15)

and a system of matrices of tolerance

4 =(01), l2

L 01

L 01

,..., L

m

Lm-1 0m-1 Lm-1 0m-1,

where 0t is the zero column with size 2t 1 x 1. It is easy to see that

(Lw v Lw ) ■ wr = cw.

(17)

and

2j < q < 2j+1( j e{1,2,., m - 2}),

then there is such element K(fx ) in the set of reduced nuclei T ( fx ) that

1) Vke{1,2.,q}k(K(f))< j +1; (11)

2) | n(K (fx )J> j +1; (12)

2) if Z2m * fX-1(1) u fX-1(-1), then either there is an element K(f% ), in the set of reduced nuclei T(fx ) which satisfies conditions (10), (11), or it is possible to construct a set of extended reduced nuclei T(fx, s), containing such an element

Proving. It is stated that function f:Gn—H2 is realized by one generalized neural element with respect to the system of characters x = {x; ,...,%,■ } of the group Gn over the field R. Let

1 m

zm=fx-1(1) u fx-1(-1),

then it follows from Theorem 1 that there exists a nucleus K ( fx ) = {a1,..., a q},

admitting representation by matrices of tolerance with Em. Consequently, there is such matrix of tolerance HeEm and such element ÇeSq, that K^(f ) = H(q). If the first row of the matrix K^(fx), is denoted by a; (£,(■) = 1), then

K ( fx ). = H,(q),

where H1 = a H e E~, follows from the equality of matrices K ( fx ) = H(q).

Consider vector w = (ra,,...,(» )eW , with coordinates

v 1 ' ' m ' m'

satisfying conditions

It follows from construction of the vector w and (16) that any matrix VeEm satisfies the condition:

Vke{1,2,.,q}k(V) < k(Ln)< j+1.

Then, on the basis of equality

K(fx), = Hi(q)(Hi eEm)

and

2' < q < 2j+1 ( j e{1,2,., ..., m- 2}), it follows that

Vk e{1,2,.,q} k(K(f) ) < j +1.

The order number of row ei in any matrix of tolerance V eEm does not exceed the order number of row e; in the matrix

Lm (i e {1,2,m -1}).

Consequently,

|n(V(q))| >|n(Lm(q))|= j +1.

follows from the inequality 2j < q < 2 j+1 and construction of the matrix Lm. In view of arbitrariness of the matrix of tolerance V eEm and K%(fx) = H1(q)(H1 eEm), we have that

|n(a!A)|>|n(L„ (q))|= j +1

and the first part of the theorem is proved when

Z 2m = fx-1(1) u fx-1(-1)

2j < q < 2 j+1 ( j e{1,2,..., m - 2}). Let

and

Z2m * fx-1(1) u fx-1(-1)

and function f:Gn®H2 be realized by one generalized neural element in relation to the system of characters x = {X^, • ■ ■, X; } of the Gn group. Then, based on Theorem 1, either the nucleus K(f% ) = {a1,w, a }, or the extended nucleus

K ( fx ,s) = {al,•, a q , aq+1.....a q+s }

admits representation by the matrices of tolerance with Em. The number of rows of an arbitrary matrix of tolerance with Em is 2m-1, therefore, q + s < 2m—1 and inequality (13) takes place for q + s. Inequalities (14), (15) are proved in the same way as the inequalities (11) and (12) were proved above. Consequently, this theorem is proved completely.

Let f:Gn®H2 be a Boolean function and f:Gn®H2 is its nucleus with respect to the system of characters

X {x i , • ■ ■ , X i }

1m

of the group Gn over the field R. s(i;K^(fx)), Denote the number of units of the i-th column of matrix K^ (fx) by s(i; ( fx )) and enter the notation K ( fx ) = K ( fx ,0).

Theorem 6. If the Boolean function f:Gn®H2 is realized by one generalized neural element with respect to the system of characters x ={x ^, • • •, X, } of the group Gn over the field R, then there is an extended nucleos

K ( fx ,s)={a1,„., aq, a q+1,..., a q+s }(s > 0)

with element

at eK(fx,s),

elements

^ e Sq+s, OeSm

than there is the following inequality for the extended reduced nucleus

K ( fx, s)t = atK ( fx, s)

and for all i e{2,3,„., m}

s(i — 1; ( fx, s)t) > s(i; (fx, s )t). (18)

Proving. According to Theorem 2, K(fx,s) admits representation by matrices of tolerance with Em, that is, there is such matrix of tolerance LeEm and such element £,e Sq, that

Kf,s) = L(q). (19)

Denote the first row of the matrix K^ (fx, s) by

at ( t) = 1)

and, transform the equality with the help of this element (19) as follows:

a ( f, S) = a L(q).

(20)

The matrix Lw = atL defines vector w eWm, all coordinates of which are negative since the first coordinate of the vector cw =(LwVLw) wT is 0. Choose the element oeSm so that the coordinates of the vector w1 = w° are arranged in a descending order. Then matrix

Lw =(atL)° = ai° L

i. e. L^ e Em . The following is obtained from (20):

aKoas) = a°L°(q) = LWi(q). (21)

Let

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a = (al,•, a i-2,0,1, a i+l,•, a „ ) be a row of the pre-matrix of tolerance Lw (q). Then

b = (a1v.v a ,—2,1, 0, ^••v a „ )

will also be a row of the matrix Lw (q) and the order number of row b in the matrix Lw (q) will be smaller than the order number a since w eW—. This means that the follow-

1n

ing inequality is fulfilled for any i e{2,3,„.,m}, and for any k e{1,2,„., q + s}:

s(i_ 1;LW1(k))> s(i;LW1(k)).

satisfies condition

(Lw1 V L; )■ w[ = cW1,

The last inequality and equality (21) directly result in inequality

s(i — 1; K- (fx, s)t) > s(i; K° ( fx, s)t) and the theorem is proved.

4. 3. Conditions sufficient for the implementation of Boolean functions by one generalized neural element

In this section, we consider conditions sufficient for re-alizeability of Boolean functions by one generalized neural element which can be successfully used in synthesis of neural networks based on the GNE with integer-valued structure vectors.

Let p be a threshold operator [23] with labels ak, ok and K(fx,s)(s > 0) is an extended nucleus of the Boolean function f:Gn®H2 with respect to the system of characters X = {X,, • • •, X, } of the group Gn over the field R. Assuming

1 m

p(K(fx ,s)) = p (a1kK °k (fx, s)), obtain

p(K ( fx ,s)) =

= pc(K( fx, s)) V p,(K( fx, s)) V^V pt0(K(fx, s)), (22) where

pc ( f) = p0 (a!kK ( fx, s)) = (la 0 ;k -0 ;k );

A(K(fx,s)) = px (a"kkK(fx,s)) = (Lj (qk)^);

pto(K(f% ,s)) = pt (alkK - (fx, s)) = (l;+t0 —1) &-.0 ),

n—( jk +to—1)

and qk > qk > • > q* —1>0.

Theorem 7. Let K(fx ,s)={a1,^, aq} is the nucleus of the Boolean function f:Gn®H2 relative to the system of characters X = {X^, • • •, X, } of the group Gn over the field R. If it is possible to construct such an extended nucleus K(fx,s)(s > 0) for which there are such elements ak e K(fx,s), £, e Sq+s and ok e Sm, that for the extended reduced nucleus K(fx, s)k = akK(fx, s), the following takes place:

K% ( fx, s)k = p (a°kK <* ( fx, s)),

(23)

then function f is realized by one generalized neural element in relation to the system of characters x .

Proving. To prove this theorem, it suffices to show that matrix

p Kk

K " ( fx ,s))

1) ra1 = -1, ra2 = ra1 - = -1

2) Coordinates ra ■ +r are found sequentially from equa-

tions:

3) % +t+1= ••• = ram =

= (z,(ra1,.,ffljk ...,ffljk+t,0,.,0))-1.

a = (a1v.v « j,..., «j+t,..., « m ) eZ

ek (a ■ ) =

a ;, if i < j -1;

a ;( j - rk X if ■ = j + k; a .j, if i > j +1;

(28)

is a tolerance prematrix of some matrix L eEm. We shall show that there exists such m-dimensional real vector

w = (ra1,...,ra m )

which satisfies the condition

Vx ea!kK°k(fx,s), Vy eZ2m \a^kK°k(fx,s)

(x,w)>(y,w) (24)

By the condition (23), matrix K^(fx,s)k admits representation (22), hence, t0 is the least positive integer such

that qtk = 0 and

t0

q%+1=.=qi- a =0.

Denote by z0,...,zt -1 the last rows of the corresponding

matrices 0

P1(K ( fx, s)),..., Pt0(K ( fx, s)), when t0 > 2, and construct vector w = (ra1v..,ram) as follows:

where

k = 0,1,..., t; r0, r1,w, rt e{1,2,., j -1}. Specify the following mapping through functions ek (k = 0,.,t) for fixed t e{0,..., m - j} and j:

ej : Z2m ^ Z™1 (Zj+1 = {0,1,., j},2 < j < m)

as follows:

ej. (a) = (ej. (a1 ),., ej (a j-1 ), e° (a ; ), ej (a;+1 ),

e (a,+t), (a,+t+l),•, e',(a m ))

(29)

and define functional vl- in the set Z2m by the formula:

(30)

Va eZ2m vj (a)= ^ ej (a s) + j (a;+!),

ieIt (j) i=0

where

It ( j) = {1,2,w, m}\{ j, j +1,., j +1}.

Using functional vl- for each k e{0,1,...,t}, construct a set of Boolean vectors as follows:

(25)

p(rk*) = rj+k

{a e h(L*+k 0.0)|0' (a) < j -1},

(31)

(zr,(<»1,...,raj ,...,ra. +r,0,...,0)) =

Jm Jk

= (zr-1,(ra1,..., ra. +r-1,0,.,0)), r = 1,., t (t = t0 -1); (26)

(27)

The vector thus constructed satisfies condition (24).

Then existence of such a vector v eW- and such a matrix

m

Lv eEm, follows from [22] that (Lv VLv)■ wT = cv. Consequently, it is possible to construct a premarix of tolerance Lv (q + s) from the elements of the extended reduced nucleus K(fx,s)k and, in accordance with Theorem 2 (in the case when t0 > 2), the function f is realized by one generalized neural element with respect to the system of characters %. If t0 =0 or t0 = 1, then, as it follows from the properties of the matrices of tolerance of the set E", the extended reduced nucleus K^ (fx, s)k admits representation by the matrices of tolerance with Em and the theorem is proved.

Use the set of vector coordinates {a1,...,a m }

to sequentially define the system of functions e°0,e1,..., ej for fixed t and j ( j > 2) in the same way as in [23] as follows:

where h(L*+k0.0) is the set of Boolean vectors which is constructed from rows of the matrix (L*+k 0.0). Let

K (fx ) = {a1,...,a q}

be nucleus of the Boolean function f:Gn®H2 relative to the system of characters

X {Xi. , • ■ ■ , Xi }

1m

of the group Gn over the field R. Similarly to Theorem 5 in [23] for generalized neural elements, we have: Theorem 8. If in the extended nucleus

K(fx,s)={a1,.,aq+s} (s>0, q+s<2m-1)

of the Boolean function f:Gn®H2 relative to the system of characters

X {Xi. , • • • , Xi }

1m

of the group Gn over the field R, there exist respectively such elements a, o and integers

r0 > r1 >... > rt > 0

in the group Sm, that

0K0(fx,s) = h(Lj 0.0) u|UF

m- j \ i

7(r, t) j+i

(32)

then function f is realized by one generalized neural element in relation to the system of characters x.

Consider generalization of the system of functions

ek (k = 0,1,2.....t)

and functional vl-. Let

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a = (ai,..v a j v.v «j+t a m) eZ2m,

t e {0,1,., m - j}, (j>2).

Construct a set of vectors u-(d ) for fixed j e{2,3,..., m} and d e {1,2, j -1}:

u j(d ) ={(u1,.,ud )|u1 +... + ud=

= j -1, U1,...,ude{1,2,., j - d}} (33)

and define the following through it:

j-1

U, = U u j (d ).

d=1

Admit that u=(u1,.,ul ) eU;, lu > 2 (lu is dimensionality of vector u) and construct a system of functions

e(u,o),e(ui),..(ui):

j j j

e( («i)

(u,k)( ai, «i 2r-1, /1..

a

Jup 2 p-1-rk +1

p=1

' lu

Zup 2p-1 +1

p=1

if i < u1,

r-1 r

if Z up<i <Z up,

p=1 p=1

if i = j + k,

if i > j + t,

(34)

where k = 0,1,., t, r e{2,3,., lu}. If lu =1, than e(u,k) = ek. For fixed

j e{2,3,., m}, t e{0,1,., m - j} and u = (u1,.,ul )eU;

specify representation

e(ut) : zm _. zm

( lu

c = Jup 2p-1 +1 p=1

v

/

as follows:

..., e(.u,i)(a .-1), e(.u,0)(a.), e^a^),.

j V j-1/> j

..., e^)(a j+t), e^ +1),., »(am))

where

It ( j) = {1,2,., m}\{ j, j +1,., j +1}. Specify a set of Boolean vectors F+Tk t) through the

functional v(u,t):

F (u, rk;) rj+k

where

h(L)+k 0.0)|v(ut )(a) <£up 2 p-1

p=1

(37)

r0,r1,.,rt e -j 1,2,.,Zup2p-1l,k e{0,1,.,t}.

p=1

Theorem 9. If in the extended nucleus

K(fx,s)={a1,.,a?+s} (s>0, q+s£2m-1)

of the Boolean function f:Gn®H2 with respect to the system of characters % = {xi ,.,%i } of the group Gn over the field R, then, respectively, there are such elements a, 0, u and such integers

i lu

r„ <Zup 2p-1

p=1

r0 > r1 > .rt >0 in the group Sm and in the set Uj that a0 K0 (f%, s ) = h(Lj 0.0) uf Uj

m- j V i

(u,r, t) j+i

(38)

then function f is realized by one generalized neural element in relation to the system of characters x.

Proving. It is given that equality (38) is valid with respect to the elements

a eK0 (fx, s), 0eSm and u = (u1,.,u^ ).

Then one can construct a vector w = (ra1,., am), satisfying the condition:

Vx e a0 K0 (fx ,s), Vy e Z2m \ a0 K0 (f, s) (x,w)>(y,w). (39)

Define coordinates ai of the vector w as follows: a1 = . = =-1,

a

1+u2+.+"/ -1 +

vi. -1

= a + + + = -2u u1+u2+.+ul

11 \ 11 \

a j = r0 - Zup 2 p-1 +1 , a j+1 = r1- Zup 2 p-1 +1

^ p=1 ) ^ p=1 )

(35)

^ l..

and define functional v(u,i) on the set Z2m with the help of the following formula:

VaeZ2m vf)(a)= Z e<"'^) + Ze(ni)(aj+i), (36)

ieIt (j) i=0

p=1

Zu 2p-1 +1 , a +t+1=. = a = -Zu 2p-1 -1.

^^ p ' j+t+1 m ^^ p

p=1

The following is obtained from the vector w cons-ruction:

a

min{(x, w) | x e h(L; 0.0)} = - ^'up 2 p-1,

p=1

Vk e {0,.,t} min{(x, w) | x e FlTk } = iu iu

= -£«p 2 p-1> up 2 p-1 -1 =

p=1

p=1

= max {(x, w )|x eh (Lj+k 0...0) \ F^},

Vz e {t +1,...,m} max{(x, w) | x e h(L*+z0• 0)} = -£up2p-1 -1.

p=1

Then (39) follows immediately from (38) and, according to Theorem 7, we can show existence of such a matrix of tolerance L1 e Em, in which the first q+s rows can be constructed from the elements of the extended nucleus aoKo(f ,s). Consequently, there exists such an element £, e Sq+s, that

ao KO (fx ,s) = A(q)

F8( u1,3) = {(0,0,0,0,0,0,0,1,0,0)}.

In accordance with Theorem 9, ra1 = ra 2 = -1, ra3 = -2, ra4 = -4, ra5 = ra6 = -6, ra7 = -7, ra 8 = -8,

ra9= ra10= -9.

Thus, if

K ( fx ,0)=fx-1(1), then a neural element with a weight vector

w1 = a swO-1 = (-9, -9, -8, -7, -6,6,4,2,1,1) and threshold [20]

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<»0=(a sxO-1,W1) = 6

and the theorem is proved.

Consider synthesis of a neural element with a generalized (x = (z,0,0,0,0,0), z is the last row of the matrix of tolerance threshold function of activation relative to the system of L5) realizes function f(x1,.,x10) with respect to the system

characters

X = {x1 = x210-1 , x2 x210-2 w, x10 = x1}.

Let n=10 as =(0,0,0,0,0,1,1,1,1,1),

1 23456789 10 10 98765432 1

j = 5, r0= r1=3, r2 = 2, rs=t r4 = r5=0

and u = (2,1,1) eU5.

Determine e(5u'3) for an arbitrary vector

a = (a1,.,a10)eZ210,

e5u,3)(a) = (e5u,3)(a1), e5u,3)(a2), e5u,3)(a3),

e5u3)(a4), e5u,0)(a5), e5u,1)(a6), e5u,2)(a7), e5u,3)(a8), e5u,3)(a9), e5u,3)(a10)) = = (a1,a2,2a3,4a4,6a5,6a6,7a7,8a8,9a9,9a10).

Construct sequentially

F(u,3,3) F(u,3,3) F(u,2,3) F(u,1,3) 5 >r6 ' 7 ' 8

according to the rule:

^+ukrk ,s) = {a e h(L)+k 0.0) | v( u's)(a) < 2 -1 +1- 2 +1- 22},

c(u,3,3) =

of characters x in alphabet {0,1}. In the opposite case, K ( fx ,0)=fx-1(-1),

the function f(x1,.,x10) is realized by a neural element with a vector of structure

[w2 =(9,9,8,7,6,-6, -4,-2,-1,-1); -5]

relative to the system of characters x in alphabet {0,1}. Remark. The operation aw of a boolean vector

a = (a1,...a m )

to a real vector

w = ram )

is defined as follows:

aw = ((-1)a1ra1,...,( - 1)a- ram ).

5. Discussion of results obtained in the study of synthesis of generalized neural elements

On the basis of the obtained necessary and sufficient conditions for realizability of the logic algebra functions by one generalized neural element in relation to the system of characters X = {X^,•••,Xi }, construct an algorithm for syn-

(1,1,0,0,1,0,0,0,0,0), (0,0,1,0,1,0,0,0,0,0)};

F5(u,3,3) = {(0,0,0,0,1,0,0,0,0,0),(1,0,0,0,1,0,0,0,0,0),(0,1,0,0,1,0,0,0,0,0), thesis of such neural elements.

Algorithm for the synthesis of generalized neural elements

Step 1. Let the Boolean function f:Gn®H2 F6(u,3,3) = {(0,0,0,0,0,1,0,0,0,0),(1,0,0,0,1,0,0,0,0,0),(0,1,0,0,0,1,0,0,0,0), and the system of characters x = {x^,---,ll } of

the group Gn be defined over the field R. Search (1,1,0,0,0,1,0,0,0,0),(0,0,1,0,0,1,0,0,0,0)};for nucleus K(fx). If the nucleus exists, then

proceed to step 2 and in the opposite case, make F5(u,2,3) = {(0,0,0,0,0,0,1,0,0,0),(1,0,0,0,0,0,1,0,0,0),(0,1,0,0,0,0,1,0,0,0)}; a conclusion that function f is not realized by

one generalized neural element relative to the system of characters

x {x i , . • •, x I } 1m

and the algorithm completes its work.

Step 2. Check the conditions necessary for realizability of the function f by one generalized neural element relative to the system x (Theorems 3-5). If the conditions of at least one of the theorems are not fulfilled, then the function f is not realized by a single GNE with respect to the system x and the algorithm completes its work, or, in the opposite case, denote the extended reduced nucleus which satisfies the conditions of all three theorems by

K (fx ,s)(s > 0)

and proceed to step 3.

Step 3. Based on the reduced nucleus K( f ,s)(s > 0), construct sequentially the elements of the set of the reduced nuclei T(fx,s), apply Theorem 6 to the constructed reduced nucleus and verify equality (23). If equality (23) is fulfilled, then, according to Theorem 7, find the vector of the structure of the generalized neuron element with respect to the system of characters x, which realizes function f in alphabet {0,1} and the synthesis of the GNE is completed. If no reduced nucleus with T (fx ,s) satisfies equality (23), then check conditions of Theorems 8 and 9 relative to

K (fx ,s)(s > 0).

If

K ( fx ,s)(s > 0)

satisfies conditions of Theorem 8, then the vector of structure of the generalized neuron element that realizes function f relative to the system x in alphabet {0,1} is to be found by Theorem 5 [23].

If K(fx,s)(s > 0) satisfies conditions of Theorem 9, then vector of the GNE structure which realizes function f in alphabet {0,1} with respect to the system x, must be found in accordance with this theorem.

If the conditions of any of the three theorems (7, 8, 9) are not fulfilled, then synthesis of the GNE for realization of function f with respect to the system x is not successful and the algorithm completes its work.

Remark 1. If the number of entries to the GNE is not limited from above (the maximum number of entries for realization of the function f:Gn®H2 in alphabet {0,1} is 2n-1), then expand the system x by adding new character(s) of the groups Gn over field R.

Example. Let f:Gn®H2 be a Boolean function,

x {x ij ,. • •, x }

is a system of characters of the group Gn over the field R (n is a natural integer satisfying the inequality 2n>8) and

f -4(1) = {(11100001), (11000001), (11110001), (11010001),

(11101001),(11001001),(11111001),

(01100001),(01000001),(10100001)}.

K ( fx ) =

aK ( fx )

Let

K( fx ) = f -1(1),

Ç be a unit element of the group

S10, a1 = (11100001).

Then

1110 0 0 0 1 110 0 0 0 0 1 11110 0 0 1 110 10 0 0 1 1110 10 0 1 110 0 10 0 1 111110 0 1 0 110 0 0 0 1 0 10 0 0 0 0 1 1 0 1 0 0 0 0 1 0000000 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 110 0 0 0 0 0 0 0 10 0 0 0 0 10 10 0 0 0 0 0 110 0 0 0 0 0 0 0 10 0 0 0 10 0 10 0 0 0 0 0 0 0 10 0 0

0 0 0 0 0 0 0 0' 0 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 110 0 0 0 0 0 0 0 10 0 0 0 0 10 10 0 0 0 0 0 110 0 0 10 0 0 0 0 0 0 10 10 0 0 0 0 0 10 0 0 0 0 0

= K0 ( fx ).

where

o =

1 2 3 4 5 6 7 8 4 5 1 2 3 6 7 8

Element o is determined by Theorem 6. The reduced nucleus K(fx)1 satisfies conditions of Theorems 3-5 and equality (23) takes place:

00000000 10000000 0 1000000 11000000 00100000 10100000 0 1100000 00010000 10010000 00001000}

K0 ( fx )1 =

4(3) 4(2)

4(1)

Therefore, K0(fx )1 =

= (L3 00000) V (L*3 (3)00000) V (l4(2)0000) V (L*5 (1)000).

Denote last rows of the blocks

(L3(3)00000), (L4(2)0000), (l5(1)000)

by

z0 = (01100000), z1 = (10010000), z2 = (00001000)

respectively, then vector of the structure of the generalized neural element in alphabet {0,1} which realizes function f:Gn®H2 with nucleus

K ( f ) = f -1(1)

in respect to the system of characters x = {X^ ,•■■,X^}, will be found in accordance with Theorem 7:

w = (ra1, ra 2, ra 3, ra 4, ra 5, ra6, ra7, ra8;ra 0 );

w1 = ra2, ra3 );

ra1 = -1, ra2 = ra1 -1 = -2, ra3 = ra1 + ra2 -1 = -4; w1 = (-1,-2, -4); W2 = (-1,-2, -4, ra 4 );

(zp w1 M^ w2 Hra = -5;

W3 = (-1, -2, -4, -5, ra5 );

(z1, w2 ) = (z2, w3 )^ra5 =-6;

w = (-1, -2, -4, -5, -6, ra6, ra7, ra 8 ) ;

ra6 = ra7 = ra8 = (z2, w 3 )-1 = -7;

w = (-1, -2, -4, -5, -6, -7, -7, -7);

w* = aw0 1 =

= (1,1,1,0,0,0,0,1) (-5, -6, -1, -2, -4, -7, -7, -7) = = (5,6,1, -2, -4, -7, -7,7).

The threshold ra0 is defined as follows:

ra0 = (a1z0 ,w*) = 13.

If K(f ) = f-1(1), then the generalized neural element with a weight vector

w* = (5,6,1, -2, -4, -7,-7,7)

and threshold ra0 = 13 realizes function fin alphabet {0,1} with respect to the system of characters

In the opposite case, that is, when

K (f)=f -1(-1),

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function f is realized by one generalized neural element in alphabet {0,1} with a weight vector

w** = - w* = (-5,-6,-1,2,4,7,7, -7) and threshold

a0 =-a0 +1 = -12.

The connection between the vectors of structure of the neural elements which realize the same function in different alphabets {0,1} and {-1,1}, was established in [20].

6. Conclusions

1. Expansion of functional capabilities of neural elements by generalization of activation functions provides for a more efficient use of these elements in the tasks of processing discrete signals and images. However, in order to successfully apply generalized neural elements in the field of compression and transmission of discrete signals, classification and recognition of discrete images, it is necessary to have practically suitable methods for checking realizability of the logic algebra functions on such elements and the methods of synthesis of these elements with a large number of entries.

2. Based on the results presented in the paper on the structure of nuclei and extended nuclei of Boolean functions with respect to the system of characters and properties of the matrices of tolerance, the following was established:

- if there is a nucleus relative to a specified system of characters for a Boolean function, then the function is realized by one generalized neural element with respect to the system of characters if and only if the nucleus or extended nucleus of the function admits representation by the matrices of tolerance;

- efficient conditions are required for checking realiz-ability of Boolean functions by one generalized neural element in relation to the system of characters;

- conditions sufficient for realizability of the logic algebra functions by one generalized neural element in respect to the system of characters on the basis of which an algorithm of synthesis of integer-valued generalized neural elements with a large number of entries has been developed.

3. The results obtained in the work can be used in working out effective methods for synthesizing neural network schemes from integer-valued generelized neural elements with a large number of entries for encoding, classification and recognition of discrete signals and images.

References

Grytsyk, V. V. Methods of parallel vertical data processing in neural networks [Text] / V. V. Grytsyk, I. G. Tsmots, O. V. Skorokho-da // Reports of the National Academy of Sciences of Ukraine. - 2014. - Vol. 10. - P. 40-44.

Tsmots, I. G. Principles of construction and methods of VLSI-implementation of real time neural networks [Text] / I. G. Tsmots, O. V. Skorokhoda, I. Ye. Vavruk // Scientific Bulletin of Ukrainian National Forestry University. - 2012. - Vol. 6. - P. 292-300. Tkachenko, R. Neiropodibni struktury mashyny heometrychnykh peretvoren u zavdanniakh intelektualnoho analizu danykh [Text] / R. Tkachenko, A. Doroshenko // Kompiuterni nauky ta informatsiini tekhnolohii. - 2009. - Vol. 638. - P. 179-184.

4. Izonin, I. V. Neural network method for change resolution of images [Text] / I. V. Izonin, R. O. Tkachenko, D. D. Peleshko, D. A. Batyuk // Information Processing Systems. - 2015. - Vol. 9 (134). - P. 30-34.

5. Marín, D. A new supervised method for blood vessel segmentation in retinal images by using gray-level and moment invariants -based features [Text] / D. Marín, A. Aquino, M. E. Gegúndez-Arias, J. M. Bravo // IEEE Transactions on Medical imaging. -2011. - Vol. 30, Issue 1. - P. 146-158. doi: 10.1109/tmi.2010.2064333

6. Azarbad, M. Automatic Recognition of Digital Communication Signal [Text] / M. Azarbad, S. Hakimi, A. Ebrahimzadeh // International Journal of Energy, Information and Communications. - 2012. - Vol. 3. Issue 4. - P. 21-33.

7. Zaychenko, Yu. P. Primenenie nechetkogo klassifikatora NEFCLASS k zadache raspoznavaniya zdaniy na sputnikovyh izobrazheni-yah sverhvysokogo razresheniya [Text] / Yu. P. Zaychenko, S. V. D'yakonova // Visnik NTUU "KPI". Informatika, upravlinnya ta obchislyuval'na tehnika. - 2011. - Vol. 54. - P. 31-35.

8. Amato, F. Artifical neural networks combined with experimental desing: a "soft" approach for chemical kinetics [Text] / F. Amato, J. L. González-Hernández, J. Havel // Talanta. - 2012. - Vol. 93. - P. 72-78. doi: 10.1016/j.talanta.2012.01.044

9. Broygham, D. Artificial neural networks for classification in metabolomic studies of whole cells using 1h nuclear magnetic resonance [Web-resource] // D. F. Brougham, G. Ivanova, M. Gottschalk, D. M. Collins, A. J. Eustace, R. O'Connor, J. J. Havel // BioMed Research International. - 2011. - P. 1-8. doi: 10.1155/2011/158094

10. Barwad, A. Artificial neural network in diagnosis of metastatic carcinoma in effusion cytology [Text] / A. Barwad, P. Dey, S. Sushei-lia // Cytometry Part B: Clinical Cytometry. - 2012. - Vol. 82B, Issue 2. - P. 107-111. doi: 10.1002/cyto.b.20632

11. Geche, F. Development of synthesis method of predictive schemes based on basic predictive models [Text] / F. Geche, O. Mulesa, S. Geche, M. Vashkeba // Technology Audit and Production Reserves. - 2015. - Vol. 3, Issue 2 (23). - P. 36-41. doi: 10.15587/23128372.2015.44932

12. Dey, P. Application of an artifical neural network in the prognosis of chronic myeloid leukemia [Text] / P. Dey, A. Lamba, S. Kumary, N. Marwaha // Analytical and quantitative cytology and histology/the International Academy of Cytology and American Society of Cytology. - 2011. - Vol. 33, Issue 6. - P. 335-339.

13. Geche, F. Development of effective time series forecasting model [Text] / F. Geche, A. Batyuk, O. Mulesa, M. Vashkeba // International Journal of Advanced Research in Computer Engineering &Technology. - 2015. - Vol. 4, Issue 12. - P. 4377-4386.

14. Liu, A. S. Automatic modulation classification based on the combination of clustering and neural network [Text] / A. S. Liu, Q. Zhu // The Journal of China Universities of Ports and Telecommunication. - 2011. - Vol. 18, Issue 4. - P. 13-38. doi: 10.1016/ s1005-8885(10)60077-5

15. Pathok, A. Data Compression of ECG Signals Using Error Back Propagation (EBP) Algorithm [Text] / A. Pathok, A. K. Wadhwa-ni // International Journal of Engineering and Advence Technology (IJEAT). - 2012. - Vol. 1. Issue 4. - P. 2249-8958.

16. Bodyanskiy, Ye. Fast training of neural networks for image compression [Text] / Ye. Bodyanskiy, P. Grimm, S. Mashtalir, V. Vi-narski // Lecture Notes in Computer Science 2010. - P. 165-173. doi: 10.1007/978-3-642-14400-4_13

17. Shovgun, N. V. Analiz effektivnosti nechetkih neyronnyh setey v zadache otsenki kreditnogo riska [Text] / N. V. Shovgun // Information technologies & knowledge. ITHEA IBS ISC. - 2013. - Vol. 7. - P. 286-293.

18. Kertis, Ch. Teoriya predstavleniy konechnyh grupp i assotsiativnyh algebr [Text] / Ch. Kertis, I. Rayner. - Moscow: Nauka, 1969. - 667 p.

19. Golubov, B. I. Ryady i preobrazovaniya Uolsha. Teoriya i primeneniya [Text] / B. I. Golubov, A. V. Efimov, V. A. Skvortsov. - Moscow: Nauka, 1987. - 343 p.

20. Dertouzos, M. Porogovaya logika [Text] / M. Dertouzos. - Moscow: Mir, 1967. - 342 p.

21. Ayzenberg, N. N. Nekotorye algebraicheskie aspekty porogovoy logiki [Text] / N. N. Ayzenberg, A. A. Bovdi, E. Y. Gergo, F. E. Geche // Kibernetika. - 1980. - Vol. 2. - P. 26-30.

22. Yadzhima, S. Nizhnyaya otsenka chisla porogovyh funktsiy. Vol. 6. Kiberniticheskiy sbornik: novaya seriya [Text] / S. Yadzhima, T. Ibaraki. - Moscow: Mir, 1969. - P. 72-81.

23. Geche, F. Verification realizibility of Boolean functions by a neural element with a threshold activation function [Text] / F. Geche, O. Mulesa, V. Buchok // Eastern-European Journal of Enterprise Technologies. - 2017. - Vol. 1, Issue 4 (85). - P. 30-40. doi: 10.15587/1729-4061.2017.90917

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