RU2318238C1 - Neuron network for transformation of residual code to binary positional code - Google Patents

Abstract

FIELD: neuron networks for transforming residual code to binary positional code, base circuit for restoring a positional number based on its residuals.

SUBSTANCE: neuron network for transformation of residual code to binary positional code contains input layer of neurons, n-neuron networks of finite ring for transformation of residual code to code of generalized positional notation system, n constant memory devices for storing binary equivalents of coefficients of generalized positional notation system and a parallel adder.

EFFECT: increased efficiency.

1 dwg

Description

The invention relates to computer technology and can be used in modular neurocomputers to implement the data output operation.

A device is known for converting a number from a system of residual classes into a positional code (AS 1005028, G06F 5/02) comprising a shift register, a synchronization unit, a constant memory unit, and a position accumulating accumulator. However, such a device is characterized by high complexity.

The closest in technical essence to the claimed device is a code converter from the system of residual classes to a binary positional code (AS 813408, G06F 5/02), containing an input register, decoders, a code converter from the system of residual classes to a polyadic code, a group of elements OR, delay elements and adder. The disadvantage of this device is the complexity of the group of OR elements when using large modules of the system of residual classes and the complexity of the converter of the residual code into a polyadic code that implements the Garner sequential algorithm.

An object of the present invention is to reduce the equipment of a residual code to binary position code converter. This goal is achieved by the fact that neural networks of the final ring and read-only memory devices are introduced into the converter. Thus, a neural network for converting a residual code into a binary positional code will consist of an input layer of neural networks acting as registers, a parallel converter of the residual code into a polyadic code, read-only memory devices and a parallel weighted adder.

, если наибольший общий делитель любой пары модулей равен 1.The inverse transformation of a number from a modular representation to binary is based on the classical theorem from number theory, which is called the Chinese remainder theorem (CTO). Based on the well-known representation of numbers in RNS (α ₁ , α ₂ , ..., α _n ) CTO makes it possible to determine the number in MSS

if the greatest common factor of any pair of modules is 1.

WHO has the form

,

для (p_i,p_j)=1 для i≠jWhere

,

for (p _i , p _j ) = 1 for i ≠ j

α _i is the remainder of the number x modulo p _i , for i = 1, 2, ..., n.

, а не само X. Если известно, что x находится между 0 и P-1, то можно записатьVarious forms of the Chinese remainder theorem (CTO) are used. From (1) it is clear that from CTO we get

, and not X itself. If it is known that x is between 0 and P-1, then we can write

In some cases, it is desirable to have the form of CTO, where the sum appears without an operator modulo P. This can be done by defining an auxiliary function R (x), so that

.Where

.

R (x) is the function x defined for any integer x.

From WHO it is clear that the expression

differs from x by a multiple of P, this difference is - P · R (x).

The function R (x) is called the rank of a number and is widely used in calculations in modular arithmetic.

и ранг числа R(x), то для перевода числа достаточно вычислить

и ввести эту сумму в диапазон [0,P] вычитанием величины, кратной P. Недостаток рассмотренного метода заключается в том, что приходится иметь дело с большими числами P_i и, кроме того, действия сложения и умножения надо выполнять в позиционной системе счисления, а полученный результат необходимо вводить в диапазон вычитаемой величины, кратной P.The rank of the number R (x) shows how many times you need to subtract the value of the range P from the resulting number in order to return it to the range. Thus, if P _i are found,

and the rank of the number R (x), then to translate the number it is enough to calculate

and enter this sum into the range [0, P] by subtracting a multiple of P. The disadvantage of this method is that you have to deal with large numbers P _i and, in addition, the addition and multiplication must be performed in a positional number system, and the result must be entered in the range of the subtracted value, a multiple of P.

являются константами выбранной системы и определяются заранее, а ранг числа R(x) - переменная величина и сложность ее вычисления линейно зависит от числа оснований СОК, что приводит к еще более сложной процедуре восстановления числа.It should be noted that P _i and

are the constants of the selected system and are determined in advance, and the rank of the number R (x) is a variable and the complexity of its calculation linearly depends on the number of bases of the RNS, which leads to an even more complicated procedure for reconstructing the number.

In addition, the use of CTO to restore the number requires complex calculations in a modular neurocomputer, since elementary processors perform operations modulo p _i , where i = 1, 2, ..., n, and not modulo P = p ₁ p ₂ . ..p _n , as required by WHO.

It can be seen from expression (2) that adders modulo P are needed to display it. This undesirable feature can be circumvented by mapping the RNS to the associated representation with a mixed base, and then to the binary representation. The coefficients a _i can be represented using n digits with mixed bases. The mapping from RNS to the generalized positional number system (OPSS) can be determined recursively using operations on small modules p _i .

To move from computations modulo P to computational moduli p _i , a method for reconstructing numbers based on the joint use of CTO and OPSS is proposed.

Let a base system p ₁ , p ₂ , ... p _{n be given} , with a range P = p ₁ p ₂ ... p _n , and orthogonal bases B ₁ , B ₂ , ..., B _n , which are defined as

where m _i are the weights of orthogonal bases.

Then WHO can be represented as

where α _i are the residues (residues) of the number X with respect to mod p _i ;

R (x) is the rank of the number.

We represent the orthogonal bases B _i in OPSS, then

where b _ij are the OPSS coefficients, i, j = 1, 2, ..., n.

Based on (7), we write X _OPSS , expression (6) in the form

Since B _i mod p _i = 0, ∀j> i, then before the first significant digit there will be i-1 zeros.

For convenience of calculations, bases can be represented in the form of a matrix

Then X _OPSS , will be written as

Wherein

where: a _i - OPSS coefficients of the number x;

α _i are the residues of the number x with respect to mod p _i ;

b _ij - orthogonal bases presented in the OPSS; i, j = 1, 2, ..., n.

When using the traditional computing base, the product α _i b _ij mod р _i can be stored in memory, and the addresses will be the remains of α _i .

If the computing base is presented in a neural network basis, then b _ij will act as the weighting coefficients of the neural network, and the remainders α _i as input signals.

The sequence of calculations for the first option has the form

. Для реализации этого метода в нейропроцессоре необходимо иметь средства для выполнения модулярных операций, например нейронные сети конечного кольца по p_i основаниям, где i=1, 2, ..., n.Two operations are required to determine all the numbers in the OPSS: one operation for retrieving from memory and one operation for summing. Compared to the well-known sequential Garner method, the gain is determined by the expression

. To implement this method in a neuroprocessor, it is necessary to have the means to perform modular operations, for example, neural networks of a finite ring on p _i bases, where i = 1, 2, ..., n.

Example. Let the bases of the system p ₁ = 3, p ₂ = 5, p ₃ = 7, p ₄ = 2. Given the number x = (2,3,0,1), presented in the RNC for the selected modules. Find the representation of this number in OPSS, that is, x = [a ₁ , a ₂ , a ₃ , a ₄ ]. Based on expression (5), we determine the orthogonal bases of the RNS: B ₁ = 70, B ₂ = 126, B ₃ = 120, B ₄ = 105. We represent the bases B _i in OPSS, then b _ij :

.

.

Due to the fact that the constants b _{ij are} determined by the choice of the system of SOK modules, then, taking into account the transfer in i-bits, they can be stored in memory, then the conversion process can be represented as

Two operations are required to determine all the numbers in the OPSS: one operation for retrieving from memory and one operation for summing. Compared to the sequential iterative process, the gain is n-1, where n is the number of RNS modules.

The obtained values of the OPSS coefficients of the number x are used to form a binary code.

A neural network for converting a residual code into a binary positional code is shown in the drawing.

The neural network contains an input layer of neurons 2, n-neural networks of the final ring (NSC) 3, n-read-only memory (ROM) 4, adder 5, the input of neural network 1, the output of neural network 6 and weight coefficients and w = b _ij 7 .

The input layer 2 is intended for temporary storage of the input residual code, that is, it acts as a register.

The NCCC converts the residual code into the OPSS code, and the number is represented in the form corresponding to the OPSS, as follows X = a ₁ + a ₂ p ₁ + a ₃ p ₁ p ₂ + ... + a _n p ₁ p ₂ ... p _n-1 , where a _i - is called the OPSS coefficient, with 0≤a _i <p _i . The weighting coefficient associated with each OPSS figure a _i is p ₁ p ₂ ... p _i-1 . Such a system has the same range of representation as the system of residual classes p = p ₁ p ₂ ... p _n . The OPSS code coefficients are memory addresses where the values of the products a _i p ₁ p ₂ ... p _i-1 are stored. The binary equivalents of the values a _i p ₁ p ₂ ... p _i-1 stored in the memory go to the input of the adder, where the binary number is restored from the residuals.

A neural network for converting the residual code into a binary positional code operates as follows.

) числа x. Коэффициенты a_i являются входными адресами ПЗУ_i 4, где хранятся двоичные эквиваленты a_ip₁p₂...p_i-1. Считанные из ПЗУ_i двоичные коды, соответствующие произведению a_ip₁p₂...p_i-1, суммируются взвешенным сумматором 5. На выходе 6 формируется двоичный код. Время преобразования кода определяется n-тактами синхронизации. Структура преобразователя, реализованная на СБИС, требует n-НСКК по модулям p_i, n-ПЗУ_i, каждое из которых хранит p_i слов разрядностью, эквивалентной числу (p_i-1)p₁p₂...p_i-1 и одного параллельного сумматора с ускоренным переносом. Сумматор может быть реализован по типу дерева, тогда время преобразования определяется выражением [log n], где n - число модулей СОК.The input number x = (α ₁ , α ₂ , ..., α _n ), represented in the system of residual classes by its residues α _i by the modules p _i (i = 1, 2, ..., n), enters the input layer 2. The input layer is associated with n NSCC. With the selected RNS modules, the structure of the neural network depends on one external parameter and adapts to it by loading the weight coefficients w = b _ij 7 defined by the expression (7) NSCC p ₁ , p ₂ , ..., p _n 3 implement the computational model (10 ), at the outputs of which the OPSS coefficients a _i (

) of the number x. The coefficients a _i are the input addresses of ROM _i 4, where the binary equivalents a _i p ₁ p ₂ ... p _i-1 are stored. The binary codes read from ROM _i corresponding to the product a _i p ₁ p ₂ ... p _i-1 are summed by the weighted adder 5. At the output 6, a binary code is generated. Code conversion time is determined by n-clock cycles. The converter structure implemented on the VLSI circuit requires n-NSCC modules p _i , n-ROM _i , each of which stores p _i words with a capacity equivalent to the number (p _i -1) p ₁ p ₂ ... p _i-1 and one parallel adder with accelerated transfer. The adder can be implemented by the type of tree, then the conversion time is determined by the expression [log n], where n is the number of RNS modules.

Claims (1)

A neural network for converting a residual code into a binary positional code containing an input layer of neurons, n-neural networks of a finite ring and an adder, characterized in that n-constant memory devices are inserted into it to store binary equivalents of a _i p ₁ p ₂ values .. .p _i-1 (i = 1, 2, ..., n), whose address inputs are outputs of n-neural networks of a finite ring that implement a computational model

, where α _i are the digits of the system of residual classes (RNS), a are the coefficients of the generalized positional number system (OPSS); b _ij - orthogonal bases presented in the OPSS; i, j = 1, 2, ..., n, p _i is the RNS module, the inputs of which are the outputs of the neurons of the input layer, and the outputs of the neural network are the outputs of the adder, the inputs of which are the outputs of n-permanent storage devices through which binary equivalent values of a _i p ₁ p ₂ ... p _i-1 .