RU2305312C2 - Neuron network for dividing numbers, represented in remainder class system - Google Patents

Abstract

FIELD: computer engineering, possible use in modular neuro-computers for fast division of numbers, represented in remainder class system.

SUBSTANCE: in accordance to the invention, neuron network contains input layer, neuron networks of end ring for determining remainder of denominator and neuron networks of end ring for determining quotient of division of two numbers.

EFFECT: increased speed of division operation, less equipment needed.

1 dwg, 1 tbl, 3 ex

Description

The invention relates to computational modular neurocomputer systems and is intended to perform the division operation on numbers represented in the system of residual classes (RNS).

A device for dividing numbers in a system of residual classes (Ovcharenko L.A., Lopatin D.S. Dividing a number in the modular code by the base of the number system // Telecommunications. - 2002. - No. 6. - P.7-10), containing tabular calculators, coherent modular code converter, display device and modulo adder.

The disadvantage of this device is the large amount of equipment and low speed division of numbers.

Closest to this invention, the technical solution is a device presented in the form of a “Neural network for rounding and scaling the numbers represented in the system of residual classes" (Decision on the grant of a patent for application No. 2003115586/09 (016529) dated 05/26/2003), containing the input a layer of neurons, a finite ring neural network (NSCC) for determining the rank of a number, a finite ring neural network for calculating the remainder at the base of n + 1, n for a finite ring neural network for calculating a scaled number.

The disadvantage of this device is the large amount of equipment and low rounding speed.

However, such neural networks are designed to round and scale the numbers represented in the system of residual classes.

The aim of this invention is to expand the capabilities of the known neural network to perform the operation of division of numbers, increase the speed of division and reduce the amount of equipment.

, нейронной сети конечного кольца 11 для выполнения финального шага при выполнении делителя остатка и выходной нейронной сети частного 7, состоящей из НСКК 8.This goal is achieved by the fact that a neural network of a finite ring is introduced into the neural network to determine the remainder of the divider. Thus, the neural network for dividing the numbers represented in the system of residual classes will consist of an input layer 2 with neurons 3, a neural network for calculating the remainder of the divider 4, consisting of neural networks of the final ring 5 by the RNS modules p ₁ , p ₂ ,. .., p _n divisible and neural networks of a finite ring 6 modulo the divider, neural networks of a finite ring 14 for computing products

, the neural network of the final ring 11 to perform the final step when performing the remainder divider and the output neural network of private 7, consisting of NSCK 8.

The structure of the neural network depends on external parameters, which are determined by the modules of the dividend, divider and private. The functioning of the neural network is determined by weighting factors, which are constants of the RNS and are determined in advance. Network training is not required.

The number A is represented in the RNS by the set of the smallest non-negative residues α ₁ , α ₂ , ..., α _n from dividing A by pairwise prime numbers p ₁ , p ₂ , ..., p _n , called modules (bases). In this case, the number A is recorded in the RNC in the following form

.Where

.

The number A lies in the range - P≤A <P, where P = p ₁ · p ₂ · ... · p _n .

Under this condition, representation (1) is one-to-one with the usual representation A in a positional number system, i.e. by (α ₁ , α ₂ , ..., α _n ) we can determine A.

The division of the number in the RNS is reduced to discarding from the dividend an appropriately selected remainder determined by the divisor. The following is the algorithm for dividing the number A by the number D with discarding the remainder under the assumption that D is either a positive integer pairwise prime with p ₁ , p ₂ , ..., p _n , or a positive integer representing the product of numbers pairwise prime with p ₁ , p ₂ , ..., p _n . The division algorithm is obtained based on the following reasoning.

, которое делится на D без остатка. Операция нахождения

заменяет операцию отбрасывания остатка от деления. При нахождении

используется алгоритм определения остатка делителя путем расширения системы оснований, где в качестве расширяемого основания используется делитель. ПустьIf A is divisible by D without remainder, then the division operation is a modular operation and is relatively simple to implement at the NSCC. Therefore, the division algorithm as an auxiliary operation includes the operation of finding the number

which is divisible by D without remainder. Find operation

replaces the operation of discarding the remainder of the division. When finding

an algorithm is used to determine the remainder of the divider by expanding the base system, where the divider is used as the expandable base. Let be

Then

.Where

.

делится без остатка на D.From the expression (3) shows that

divisible by D.

Consider the algorithm for determining the remainder of the divider.

. Добавим к числу оснований СОК делитель D=р_n+1. Диапазон этой системы равен

. Тогда любое число А из диапазона [0, P_n+1) в обобщенной позиционной системе счисления (ОПСС) представим как

, где а_i - коэффициенты обобщенной позиционной системы счисления.Let the RNS consists of the bases p ₁ , p ₂ , ..., p _n . The range of numbers of this system will be

. Add to the number of bases JUICE divisor D = p _{n + 1} . The range of this system is

. Then any number A from the range [0, P _{n + 1} ) in the generalized positional number system (OPSS) can be represented as

, where a _i are the coefficients of the generalized positional number system.

If the number A will lie in the initial range [0, P _n ), then in the generalized positional system the figure is a _{n + 1} = 0. This fact is used to obtain the remainder of the divisor by dividing the number A by the new base RNS p _{n + 1} , which is identified with the divisor.

в системе оснований р₁,р₂,...,р_n, D=p_n+1, где

- остаток от деления числа А на р_n+1, т.е. искомая цифра по новому основанию, равному делителю D.Let the number A have a representation (α ₁ , α ₂ , ..., α _n ) on the bases p ₁ , p ₂ , ..., p _n . Add a new base p _{n + 1} = D, then the number

in the base system p ₁ , p ₂ , ..., p _n , D = p _{n + 1} , where

- the remainder of dividing the number A by p _{n + 1} , i.e. the desired figure on a new basis equal to the divisor D.

в проводимые операции. При этом мы параллельно получаем цифры в ОПСС a₁,a₂,...,а_n и выражение для цифры a_n+1=0. Из полученных соотношений и определяем остаток числа по модулю делителя

.To determine this figure, the method of converting a number from RNS to OPSS, including an unknown figure, is used.

in ongoing operations. At the same time, we simultaneously obtain the numbers in OPSS a ₁ , a ₂ , ..., and _n and the expression for the figure a _{n + 1} = 0. From the obtained relations, we determine the remainder of the number modulo the divider

.

We expand the number A in the base system p ₁ , p ₂ , ..., p _n , p _{n + 1} , then

- диапазон расширенной системы оснований,Where

- range of extended base system,

- ортогональные базисы расширенной системы оснований.

- orthogonal bases of the extended base system.

в обобщенной позиционной системе счисления, тогдаWe represent orthogonal bases

in a generalized positional number system, then

- коэффициенты ОПСС; i,j=1,2,...,n.Where

- OPSS coefficients; i, j = 1,2, ..., n.

Based on (5), we write expression (4) in the form

From the expression (6), we can determine the coefficients of the generalized positional number system a _{i of the} number A, then

where α _i are the residues of the number A by p _i ;

- ортогональные базисы, представленные в ОПСС.

- orthogonal bases presented in the OPSS.

и переносом, генерируемым при формировании a_i-1. Перенос генерируется как число раз, когда сумма цифр в ОПСС переполняется по модулю p_i. Этот перенос используется для формирования цифр a_i+1. Последний перенос, генерируемый при получении последней цифры числа в ОПСС, отбрасывается. Рассмотренный метод выполняется в параллельном режиме. Цифры α_i,

принимают от 0 до p_i-1, причем

являются константами, поэтому произведение

можно поместить в ПЗУ или в весовые коэффициенты связей между нейронами. Адресами произведений

являются вычеты α_i числа А по модулю p_i.The number a _i in the OPSS representation is obtained by summing modulo p _i all the products

and the transfer generated by the formation of a _i-1 . The transfer is generated as the number of times when the sum of the digits in the OPSS is overflow modulo p _i . This transfer is used to form the numbers a _{i + 1} . The last transfer generated when the last digit of the number is received in the OPSS is discarded. The considered method is executed in parallel. The numbers α _i ,

take from 0 to p _i-1 , and

are constants, therefore the product

can be placed in ROM or in weights of connections between neurons. Addresses of works

are the residues α _{i of the} number A modulo p _i .

To illustrate the definition of constants, we give an example 1.

Example 1. Let p ₁ = 2, p ₂ = 3, p ₃ = 5, D = 7, P _{n + 1} = 2 · 3 · 5 · 7 = 210, B ₁ = 105, B ₂ = 70, B ₃ = 126, B ₄ = 120.

:Further, on the basis of (5) we define

:

Example 2. Let a system of modules p ₁ = 2, p ₂ = 3, p ₃ = 5 be given, then P _n = 2 · 3 · 5 = 30. And let the number X = 23 = (1, 2, 3) be given. We expand the base system, where as p _{n + 1} we take the divisor D = 7. It is necessary to determine the balance on the basis of an equal divisor.

Let X = 23 = (1, 2, 3, | A | ₇ ) in the base system p ₁ = 2, p ₂ = 3, p ₃ = 5, p ₄ = D = 7.

, приведенными в (8), которые заданы матрицейWe use constants

given in (8), which are given by the matrix

The process of solving the problem is shown in table 1.

Table 1 The residue of the number X modulo p _i Modules p ₁ = 2 p ₂ = 3 p ₃ = 5 p ₄ = 7 one one one 2 3 2 0 four 2 four 3 0 0 3 12 | A | ₇ 0 0 0 4 | A | ₇ OPSS coefficients of the number A one 2 3 6 + 4 | A | ₇

, но по условию a₄=0, т.е. 4|A|₇=-6 или

. Мультипликативная обратная величина

, и так как число 6 отрицательное, возьмем его дополнение по модулю 7.As

, but by condition a ₄ = 0, i.e. 4 | A | ₇ = -6 or

. Multiplicative Inverse

, and since the number 6 is negative, we take its complement modulo 7.

So, the residue of the number A modulo p ₄ = D = 7 is determined by the expression | A | ₇ = 2 · (7-6) = 2.

Since the result of the formation of a digit in the RNS on a new basis p _{n + 1} = D depends only on the first digits, the operation of determining the residues can be carried out immediately on several divisors, pairwise simple with the bases of the RNS.

The advantage of the proposed method for determining the residues of the initial number by several divisors is that:

- all calculations are performed in parallel channels on separate modules;

и мультипликативных величин по расширенным основаниям;- calculation of a large number of additional quantities is not required, only the presence of constants is necessary

and multiplicative values on extended grounds;

- it is possible to obtain an extended representation of the deductions of the number at once for several additional reasons, which does not affect the performance of the entire expansion operation.

Based on the extensions, the algorithm for dividing the number A by the number D with discarding the remainder can be represented as a sequence of the following operations:

determination of α _{n + 1} based on a computational model (7);

на основе вычислительной модели (3);calculation

based on a computational model (3);

на Dfinding quotient

on D

This operation is a collection of modular operations that implement the expression

If p _i are prime numbers

The values of d ₁ , d ₂ , ..., d _n are independent of A and they are calculated in advance from the given D, i.e. the numbers d ₁ , d ₂ , ..., d _n are operation constants. The algorithm for dividing the number A by the number D, which is the product of numbers pairwise prime with p ₁ , p ₂ , ..., p _n , can be represented as a sequence of algorithms for dividing the number by numbers pairwise simple with p ₁ , p ₂ , .. ., p _n .

Example 3. Let a base system be given, as in Example 1. It is required to divide the number A = (1, 2, 3) by D = 7.

In example 2, the remainder is calculated modulo D and it is equal to α ₄ = 2.

по формуле (3)After that they are calculated

by the formula (3)

Next, we calculate by the formula (11) d ₁ = | 7 ⁰ | ₂ = 1, d ₂ = | 7 ¹ | ₃ = 1 and d ₃ = | 7 ³ | ₅ = 3.

Finally, using formula (10), we calculate the quotient F = (| 1 · 1 | ₂ , | 1 · 0 | ₃ , | 1 · 3 | ₅ ) = (1, 0, 3).

It is easy to verify that F = 23: 7≈3 → (1, 0, 3).

The proposed division algorithm differs from the known one in that its implementation consists entirely of modular operations modulo p _i and it can be easily implemented by neural networks of a finite ring.

The drawing shows a diagram of a neural network for dividing the numbers represented in the RNS.

The principle of operation of the present invention is described below.

The neural network shown in the drawing allows the operations of dividing the original number (α ₁ , α ₂ , ..., α _n ) 1 by the number D corresponding to the expandable module p _{n + 1} .

The remains of the dividend (α ₁ , α ₂ , ..., α _n ) 1 according to the base system p ₁ , p ₂ , ..., p _n are fed to the input of neurons 3 of the input layer 2, the remainder of the divider, taking into account transfers 10, is formed by the neural network 4, consisting of NSCC 5, 6, 14, the final step of which is calculated by NSCC 11, and the result (partial) in the residues appears at the output of the neural network of private 7, outputs 9.

. НСКК 5, 6 вычисляют значение

с учетом переносов 10. Выходные значения НСКК 5, 6 с весовыми коэффициентами 13, равными 1, подаются на вход НСКК 14 для выполнения суммирования значений

.From the outputs of neurons 3 of the input layer 2, the remainder of the dividend (α ₁ , α ₂ , ..., α _n ) through the modules p ₁ , p ₂ , ..., p _n goes to the input of the neural network 4, consisting of NSCC 5, 6 , according to the relevant modules. Weighting factors of 12 neurons NSCC 5, 6, performing the role of distributed memory, are determined by the values

. NSCC 5, 6 calculate the value

taking into account transfers 10. Output values of NSCC 5, 6 with weighting factors 13 equal to 1 are fed to the input of NSCC 14 for summing the values

.

. На выходе НСКК 11 формируется остаток делителя, равный

, который поступает на первые входы нейронной сети 7 НСКК 8. На вторые входы НСКК 8 с входного слоя поступают остатки делимого α₁,α₂,...,α_n. Весовые коэффициенты 16 НСКК 8 определяются выражением

, где i=1,2,...,n. Используя распределительный закон алгебры, выходные НСКК 8 реализуют вычислительную модель

, на выходах которой формируется частное 9.The summation results from the output of NSCC 14 modulo D (the divider) corresponding to the remainder of the divider are supplied in an additional code to the input of the NSCC 11 neural network to perform the final step, the weight coefficient of which 15 is determined by the multiplicative value

. At the output of NSCC 11, the remainder of the divider is formed, equal to

, which enters the first inputs of the neural network 7 of the NSCC 8. The second inputs of the NSCC 8 from the input layer receive the remains of the divisible α ₁ , α ₂ , ..., α _n . Weighting factors 16 NSCC 8 are determined by the expression

, where i = 1,2, ..., n. Using the distribution law of algebra, the output of NSCC 8 implements a computational model

at the outputs of which a quotient 9 is formed.

The number division time is determined by two synchronization cycles: one cycle to form the remainder of the divider and one cycle to form the quotient.

The work was performed under grant A 04-2.8-755.

Claims (1)

A neural network for dividing the numbers represented in the system of residual classes, containing an input layer of neurons whose outputs are branched to the inputs of neural networks of a finite ring that are part of the output neural network of a private and to the inputs of neural networks of a finite ring by modules p ₁ , p ₂ ,. .. _{p n} , p _{n + 1} included in the neural network to calculate the remainder of the divider, which implements the computational model α _{n + 1} = | A | _D , where A is the initial number represented in the system of residual classes, D is the divider, D = p _{n + 1} , characterized in that a neural network of a finite ring is introduced into the neural network to calculate the remainder of the divider to calculate the sum of the products

where α _i are the residues of the original number A modulo p _i ,

- orthogonal bases represented in the generalized positional number system, i, j = 1,2, ... n, and the neural network of a finite ring with a weight coefficient equal to the inverse multiplicative quantity

to perform the final step in calculating the remainder of the divider, the input of which is connected to the inverse output of the neural network of the final ring to calculate the sum of the products

modulo divider D, to the input of which values

from the outputs of the neural networks of the final ring by modules p ₁ , p ₂ , ... p _n , p _{n + 1} , the output of the neural network of the final ring to perform the final step when calculating the remainder of the divider is connected to the first inputs of the neural networks of the final ring included in the composition of the output neural network of the private, the second inputs of which receive the remains of the original number in modules p ₁ , p ₂ , ... p _n , which implements a computational model

where

and where d _i are operation constants calculated in advance by a given divisor D and not depending on the initial number A, on the output neural network of the quotient, the result is obtained from dividing the initial number by the given divider.