=Paper=
{{Paper
|id=Vol-2588/paper9
|storemode=property
|title=Methods of nulling numbers in the system of residual classes
|pdfUrl=https://ceur-ws.org/Vol-2588/paper9.pdf
|volume=Vol-2588
|authors=Victor Krasnobayev,Alina Yanko,Oleksii Smirnov,Tetiana Kuznetsova
|dblpUrl=https://dblp.org/rec/conf/cmigin/KrasnobayevYSK19
}}
==Methods of nulling numbers in the system of residual classes==
https://ceur-ws.org/Vol-2588/paper9.pdf
Methods of nulling numbers in the system of
residual classes
Victor Krasnobayev 1[0000-0001-5192-9918], Alina Yanko 2[0000-0003-2876-9316],
Oleksii Smirnov 3[0000-0001-9543-874X] and Tetiana Kuznetsova 1[0000-0001-6154-7139]
1
V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine
v.a.krasnobaev@gmail.com, kuznetsova.tatiana17@gmail.com
2
Poltava National Technical Yuri Kondratyuk University, Poltava, Ukraine
al9_yanko@ukr.net
3
Central Ukrainian National Technical University, avenue University, 8, Kropivnitskiy, 25006,
Ukraine, dr.smirnovoa@gmail.com
Abstract. The article presents the nullification of numbers in the system of re-
sidual classes (SRC). This method is widely used in the non-positional number
system in the SRC with the need to determine positional characteristics. Two
methods of nullification are presented in the article: the method of successive
subtractions and the method of parallel subtractions. Based on these methods,
algorithms are developed for their implementation. The essence of the method
of successive subtractions is that the nullification procedure is carried out con-
sequently from the junior foundation to the oldest. The essence of the parallel
subtraction method is that the nullification procedure is carried out parallel in
time for two reasons. It is advisable to use these methods in the implementation
of operations of comparing numbers in the SRC, and in monitoring data pre-
sented in the SRC. The estimation of these methods by the number of equip-
ment and by the time of implementation of the nullification procedure is made.
In terms of the efficiency of the nullification procedure, a parallel subtraction
method was proposed.
Keywords: methods of nulling numbers; method of successive subtractions;
nullification constants; nullification block; nullification procedure; parallel nul-
lification method; sequential nullification; system of residual classes.
1 Methods of data control in the system of residual classes
based on the principle of nulling
One of the methods for determining the correctness of a number is the nulling (nulli-
fication) method, which consists in the transition from the initial number.
A (a1 , a2 , ..., an , an 1 ) (1)
to the number:
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attrib-
ution 4.0 International (CC BY 4.0) CMiGIN-2019: International Workshop on Conflict Management in
Global Information Networks.
AZ (0, 0, ..., 0, n 1 ) (2)
with the help of such a sequence of transformations in which there is no output out-
side the working range of the system of residual classes (SRC) [1-8].
The process of nulling a number consists in successively subtracting from this
number the nulling constants of the form:
(m1.1 , m1.2 , ..., mn 1 ) , where m1.1 (1, 2, ..., m1 1) ;
(0, m2.2 , ..., mn 1,2 ) , where m2.2 (1, 2, ..., m2 1) .
In this case, the number A (a1 , a2 , ..., an , an 1 ) is sequentially converted to a form
A (0, a2 , ..., an , an 1 ) , then to a number A (0, 0, ..., an, an1 ) etc.
By repeating the process n times, we get:
A( Z ) (0, 0, ..., 0, n 1 ) .
If n 1 0 , then the original number is correct and lies in the range [0, M ) , if
n 1 0 , then the number is incorrect and lies in the range [ jM , ( j 1) M ) , for
j (1, 2, ..., mn 1 1) , where ( j 1) is the value of the number of the interval in
which the operand A falls.
We show that an error A can transfer the correct number A , lying in the interval
[0, M ) , only in one of the two intervals.
We write A A A , i.e.
A (a1 , a2 , ..., an , an1 ) (0, 0, ..., ai , ..., 1) . (3)
Obviously, A is not in the first (working) interval [0, M ) , since the first interval
contains the correct number:
A0 (0, 0, ..., 0, 0) .
Let A be in the k -th interval:
(k 1)M A kM .
We write the system of two inequalities:
0 A M ,
(k 1) M A kM .
Add up the two inequalities:
(k 1)M A A (k 1)M .
Let j k 1 , then we can write:
jM A ( j 2)M ,
i.e. an error may cause the correct operand to be incorrect, lying only in one of two
intervals [ jM , ( j 1) M ) or [( j 1)M , ( j 2)M ) .
2 Method of successive subtractions
Consider the nulling procedure in terms of the time of its implementation. The es-
sence of the first (Z1), the basic in the theory of SRC, nullification procedure (the
procedure of sequential nullification (SN)) consists of a sequence of operations for
subtracting in form:
A(i 1) A(i ) NC (i ) , (4)
by means of a set of NC of the form (5):
NC (0) [t1(0) || t2(0) || t3(0) || || ti(0)
1 || ti
(0)
|| ti(0)
1 || || tn(0)3 || tn(0) 2 || tn(0)1 || tn(0) || tn(0)1 ] ,
t1(0) a1(0) , t1 0, m1 1 ;
(0)
NC (1) [0 || t2(1) || t3(1) || || ti(1) (1)
1 || ti || ti(1)
1 || || tn(1)3 || tn(1) 2 || tn(1)1 || tn(1) || tn(1)1 ] ,
t2(1) a2(1) , t2(1) 0, m2 1 ;
NC (2) [0 || 0 || t3(2) || || ti(2) (2)
1 || ti || ti(2)
1 || || tn(2)3 || tn(2) 2 || tn(2)1 || tn(2) || tn(2)1 ] ,
t3(2) a3(2) , t3(2) 0, m3 1 ;
NC (i 1) [0 || 0 || 0 || 0 || ti(i 1) || ti(i 11) || || tn( i31) || tn( i21) || tn( i11) || tn( i 1) || tn( i11) ] ,
ti( i 1) ai( i 1) , ti(i 1) 0, mi 1 ;
NC ( n 2) [0 || 0 || 0 || || 0 || 0 || 0 || || 0 || 0 || tn( n1 2) || tn( n 2) || tn( n1 2) ] ,
tn( n1 2) an( n1 2) , tn( n1 2) 0, mn 1 1 ;
NC ( n 1) [0 || 0 || 0 || || 0 || 0 || 0 || || 0 || 0 || 0 || tn( n 1) || tn( n11) ] ,
tn( n 1) an( n 1) , tn( n 1) 0, mn 1 , (5)
from the corresponding numbers:
A(0) [a1(0) || a2(0) || a3(0) || || ai(0) (0)
1 || ai || ai(0) (0) (0) (0) (0) (0)
1 || ... || an 3 || an 2 || an 1 || an || an 1 ] ,
A(1) [0 || a2(1) || a3(1) || || ai(1)1 || ai(1) || ai(1)1 || || an(1)3 || an(1) 2 || an(1)1 || an(1) || an(1)1 ] ,
A(2) [0 || 0 || a3(2) || || ai(2) (2)
1 || ai || ai(2)
1 || || an(2)3 || an(2) 2 || an(2)1 || an(2) || an(2)1 ] ,
A(i 1) [0 || 0 || 0 || || 0 || ai(i 1) || ai(i 11) || || an(i31) || an(i21) || an( i11) || an( i 1) || an( i11) .
For example: performing the first subtraction operation:
A(1) A(0) NC (0) [a1(0) || a2(0) || a3(0) || ... || ai(0) 1 || ai
(0)
|| ai(0)
1 ||
... || an(0)3 || an(0) 2 || an(0)1 || an(0) || an(0)1 ] [t1(0) || t2(0) || t3(0) || || ti(0) (0)
1 || ti || ti(0)
1 ||
3 || tn 2 || tn 1 || tn || tn 1 ] [a1
... || tn(0) t1(0) ]mod m1 || [a2(0) t2(0) ]mod m2
(0) (0) (0) (0) (0)
|| [a3(0) t3(0) ]mod m3 || ... [ai(0)
1 ti 1 ]mod mi 1 || [ ai
(0) (0)
ti(0) ]mod mi
1 ti 1 ]mod mi 1 || ... [ an 3 tn 3 ]mod mn 3 || [ an 2 t n 2 ]mod mn 2
|| [ai(0) (0) (0) (0) (0) (0)
1 ]mod mn 1 [an tn ]mod mn || [an 1 tn 1 ]mod mn 1
|| [an(0)1 tn(0) (0) (0) (0) (0)
[0 || a2(1) || a3(1) || || ai(1)1 || ai(1) || ai(1)1 || || an(1)3 || an(1) 2 || an(1)1 || an(1) || an(1)1 ] ;
performing the second subtraction:
A(2) A(1) NC(1)
[0 || a2(1) || a3(1) || || ai(1)1 || ai(1) || ai(1)1 || || an(1)3 || an(1) 2 || an(1)1 || an(1) || an(1)1 ]
[0 || t2(1) || t3(1) || || ti(1) (1)
1 || ti || ti(1)
1 || || tn(1)3 || tn(1) 2 || tn(1)1 || tn(1) || tn(1)1 ]
0 || [a2(1) t2(1) ]mod m2 [a3(1) t3(1) ]mod m3 || ... [ai(1)
1 ti 1 ]mod mi 1 ||
(1)
|| [ai(1) ti(1) ]mod mi [ai(1)1 ti(1)
1 ]mod mi 1 || ... [ an 3 t n 3 ]mod mn 3 ||
(1) (1)
|| [an(1) 2 tn(1) 2 ]mod mn 2 [an(1)1 tn(1)1 ]mod mn 1 || [ an(1) tn(1) ]mod mn
|| [an(1)1 tn(1)1 ]mod mn 1
[0 || 0 || a3(2) || || ai(2) (2)
1 || ai || ai(2)
1 || || an(2)3 || an(2) 2 || an(2)1 || an(2) || an(2)1 ] ;
performing the third subtraction:
A(3) A(2) NC(2)
[0 || 0 || a (2)
3 || || a(2)
i 1 1 || ... || an 3 || an 2 || an 1 || an || an 1 ]
|| ai(2) || ai(2) (2) (2) (2) (2) (2)
[0 || 0 || t3(2) || || ti(2)
1 || ti
(2)
|| ti(2)
1 || || tn(2)3 || tn(2) 2 || tn(2)1 || tn(2) || tn(2)1 ]
0 || 0 [a3(2) t3(2) ]mod m3 || ... [ai(2)
1 ti 1 ]mod mi 1 [ai
(2) (2)
ti(2) ]mod mi
1 ti 1 ]mod mi 1 || ... [ an 3 tn 3 ]mod mn 3 [ an 2 t n 2 ]mod mn 2
|| [ai(2) (2) (2) (2) (2) (2)
1 tn 1 ]mod mn 1 || [an tn ]mod mn
|| [an(2) 1 ]mod mn 1
[an(2)1 tn(2)
(2) (2) (2)
[0 || 0 || 0 || a4(3) || a5(3) || ... || ai(3) (3)
1 || ai || ai(3)
1 || || an(3)3 || an(3) 2 || an(3)1 an(3) || an(3)1 ] ,
etc. The algorithm for performing the SN procedure is presented in Table. 1. In ac-
cordance with this algorithm, the initial number A A(0) (a1(0) || a2(0) || ... || ai(0) ||
|| ai(0) (0) (0)
1 || ... || an || an 1 ) according to the formula (4) is sequentially converted to the
form A( Z ) (0 || 0 || ... || 0 || n 1 ) with the help of a sequence of operations that does
not result in the output of the numerical value of the number А(0) over the working
range [0, M) of the SRC. In this case, the initial number
A A(0) (a1(0) || a2(0) || ... || ai(0) || ai(0) (0) (0)
1 || ... || an || an 1 ) is sequentially reduced to the
form А(H), i.e.
A A(0) (a1(0) , a2(0) ,..., ai(0) , ai(0) (0) (0)
1 ,..., an , an 1 ) ,
A(1) (0, a2(1) , a3(1) ,..., an(1) , an(1)1 ) ,
A(2) (0, 0, a3(2) ,..., an(2) , an(2)1 ) ,
A(3) (0, 0, 0, a4(3) ,..., an(3) , an(3)1 ) and so on.
Table 1. SN Algorithm
Operation
Contents of operation
number
(0) (0)
1 Appeal by value a1 and number A(0) in BNC0 for the NC .
2 Performing a subtraction operation A
(1)
A NC .(0) (0)
(1) (1) (1)
3 Appeal by value a 2 and number A in BNC1 for the NC .
4 Performing a subtraction operation A
(2)
A(1) NC(1) .
( 2) (2)
5 Appeal by value a3 and number A(2) in BNC2 for the NC .
6 Performing a subtraction operation A
(3)
A NC .(2) (2)
(3) (3) (3)
7 Appeal by value a 4 and number A in BNC3 for the NC .
8 Performing a subtraction operation A
(4)
A NC .(3) (3)
…
( n 2) ( n 2)
2n 3 Appeal by value an 1 and number A( n 2) in BNCn 2 for the NC .
2n 2 Performing a subtraction operation A
( n 1)
A ( n 2)
NC ( n 2)
.
( n 1) ( n 1)
2n 1 Appeal by value an and number A( n 1) in BNCn 1 for the NC .
( n 1) ( n 1)
Performing a subtraction operation A
( n)
A NC . Getting a nulli-
2n fied number
A( Z ) A( n ) [0 || 0 || ... || 0 || ... || 0 || 0 || n 1 an( n)1 ] .
By repeating the subtraction n times, we get the value A( Z ) (0 || 0 || ... || 0 || an( n)1 ) , or
A( Z ) (0 || 0 || ... || 0 || n 1 ) , where n 1 an( n)1 . The SN procedure is shown in Fig. 1.
Opera-
tion
Contents of operation
(cycle)
number
Appeal by the value of the remainder a1(0) of a number
A A (0)
[a (0)
1 || a (0)
2 ||
(0) (0) (0)
1 || a3 || || a i 1 || a i || ai(0) (0) (0) (0) (0) (0)
1 || ... || an 3 || an 2 || an 1 || an || an 1 ] in BNC0
for the NC NC (0)
[t (0)
1 || t (0)
2 || t (0)
3 || ... || t (0)
i 1 || t i
(0)
|| t (0)
i 1 || ...
|| t (0)
n 3 || t (0)
n2 || t (0)
n 1 || t (0)
n || t (0)
n 1 ]; t (0)
1 a ; t (0)
1
(0)
1 0, m1 1 .
Performing a subtraction operation A(1) A(0) NC (0) [a1(0) || a2(0) ||
|| a3(0) || || ai(0) 1 || ... || an 3 || an 2 || an 1 || an || an 1 ]
(0)
1 || ai || ai(0) (0) (0) (0) (0) (0)
2
[t1(0) || t2(0) || t3(0) || || ti(0)
1 || ti
(0)
|| ti(0)
1 || || tn(0)3 || tn(0) 2 || tn(0)1 || tn(0) || tn(0)1 ]
[a1(0) t1(0) ]mod m1 ||[a2(0) t2(0) ]mod m2 [a3(0) t3(0) ]mod m3 || ...
1 ti 1 ]mod mi 1 || [ ai
... [ai(0) ti(0) ]mod mi || [ai(0)
1 ti 1 ]mod mi 1 || ...
(0) (0) (0)
... [an(0)3 tn(0)3 ]mod mn 3 || [an(0) 2 tn(0) 2 ]mod mn 2 || [an(0)1 tn(0)1 ]mod mn 1 ||
|| [an(0) tn(0) ]mod mn || [an(0)1 tn(0)1 ]mod mn 1 [0 || a2(1) || a3(1) || || ai(1) (1)
1 || ai ||
|| ai(1)1 || ... || an(1)3 || an(1) 2 || an(1)1 || an(1) || an(1)1 ] .
Appeal by the value of the remainder a2(1) of a number
A (1)
[0 || a (1)
2 || a (1)
3 ||
(1) (1) (1)
3 ... || a i 1 || a i || a i 1 || || an(1)3 || an(1) 2 || an(1)1 || an(1) || an(1)1 ] in BNC1 for the
NC NC (1) [0 || t2(1) || t3(1) || || ti(1) (1)
1 || ti || ti(1) (1) (1)
1 || ... || t n 3 || t n 2 ||
|| tn(1)1 || tn(1) || tn(1)1 ] ; t2(1) a2(1) ; t2(1) 0, m2 1 .
Performing a subtraction operation
A(2) A(1) NC (1) 0 || a2(1) || a3(1) ||
... || ai(1) (1)
1 || ai || ai(1) (1) (1) (1) (1) (1)
1 || ... || an 3 || an 2 || an 1 || an || an 1 0 || t2 || t3 || ... || ti 1 ||
(1) (1) (1)
4 ti(1) || ti(1)
1 || || tn(1)3 || tn(1)2 || tn(1)1 || tn(1) || tn(1)1 0 || [a2(1) t2(1) ]mod m2
|| [a3(1) t3(1) ]mod m3 || ... [ai(1)1 ti(1)
1 ]mod mi 1 [ ai ti ]mod mi
(1) (1)
[ai(1)1 ti(1)
1 ]mod mi 1 || ... [ an 3 tn 3 ]mod mn 3 [ an 2 tn 2 ]mod mn 2
(1) (1) (1) (1)
|| [an(1)1 tn(1)1 ]mod mn 1 || [an(1) tn(1) ]mod mn [an(1)1 tn(1)1 ]mod mn 1
[0 || 0 || a3(2) || || ai(2) (2)
1 || ai || ai(2)
1 || || an(2)3 || an(2) 2 || an(2)1 || an(2) || an(2)1 ] .
Appeal by the value of the remainder a3( 2) of a number
A (2)
[0 || 0 || a (2)
3 ||
(2) (2) (2)
5 ... || a i 1 || a i || a i 1 || || an(2)3 || an(2) 2 || an(2)1 || an(2) || an(2)1 ] in BNC2 for the NC
NC (2) [0 || 0 || t3(2) || || ti(2) (2)
1 || ti || ti(2) (2) (2) (2)
1 || ... || t n 3 || t n 2 || t n 1 ||
|| tn(2) || tn(2)1 ] ; t3(2) a3(2) ; t3(2) 0, m3 1 .
Performing a subtraction operation A(3) A(2) NC (2) [0 || 0 || a3(2) ||
... || ai(2) 1 || ... || an 3 || an 2 || an 1 || an || an 1 ] [0 || 0 || t3 || ... || ti 1 ||
(2)
1 || ai || ai(2) (2) (2) (2) (2) (2) (2) (2)
|| ti(2) || ti(2) 3 || tn 2 || tn 1 || tn || tn 1 ] 0 || 0 [a3 t3 ]mod m3 || ...
|| tn(2) (2) (2) (2) (2) (2) (2)
1 ||
6
1 ti 1 ]mod mi 1 [ ai
... [ai(2) ti(2) ]mod mi [ ai(2)
1 ti 1 ]mod mi 1 || ...
(2) (2) (2)
... [an(2)3 tn(2)3 ]mod mn 3 [an(2) 2 tn(2) 2 ]mod mn 2 [an(2)1 tn(2)1 ]mod mn 1 ||
1 tn 1 ]mod mn 1
|| [an(2) tn(2) ]mod mn [an(2) (2)
[0 || 0 || 0 || a4(3) || a5(3) || || ai(3) (3)
1 || ai || ai(3)
1 || || an(3)3 || an(3) 2 || an(3)1 || an(3) || an(3)1 ]
.
Appeal by the value of the remainder a4(3) of a number
A (3)
[0 || 0 || 0 || a (3)
4 ||
(3) (3) (3)
7 || a 5 || || a i 1 || a i || ai(3) (3) (3) (3) (3) (3)
1 || ... || an 3 || an 2 || an 1 || an || an 1 ] in BNC3 for
the NC NC (3) [0 || 0 || 0 t4(3) || t5(3) || || ti(3) (3)
1 || ti || ti(3)
1 ||
|| tn(3) 3 || tn(3) 2 || tn(3)1 || tn(3) || tn(3)1 ] ; t4(3) a4(3) ; t4(3) 0, m4 1 .
Performing a subtraction operation A(4) A(3) NC (3) [0 || 0 || 0 a4(3) ||
|| a5(3) || || ai(3) 1 || ... || an 3 || an 2 || an 1 || an || an 1 ] [0 || 0 || 0 t 4 ||
(3)
1 || ai || ai(3) (3) (3) (3) (3) (3) (3)
|| t5(3) || || ti(3) (3)
1 || ti || ti(3)
1 || || tn(3)3 || tn(3) 2 || tn(3)1 || tn(3) || tn(3)1 ]
8 0 || 0 0 [a4(3) t4(3) ]mod m4 || [a5(3) t5(3) ]mod m5 || ... [ai(3)
1 ti 1 ]mod mi 1
(3)
[ai(3) ti(3) ]mod mi [ai(3)
1 ti 1 ]mod mi 1 || ... [ an 3 t n 3 ]mod mn 3
(3) (3) (3)
|| [an(3) 2 tn(3) 2 ]mod mn 2 [an(3)1 tn(3)1 ]mod mn 1 || [an(3) tn(3) ]mod mn
|| [an(3)1 tn(3)1 ]mod mn 1 [0 || 0 || 0 || 0 || a5(4) || || ai(4) (4)
1 || ai || ai(4)
1 ||
... an(4)3 || an(4) 2 || an(4)1 an(4) || an(4)1 ] .
Appeal by the value of the remainder ai( i 1) of a number
( i 1)
A [0 || 0 || 0 || || 0 ||
( i 1) ( i 1) ( i 1)
|| a i || a i 1 || || a n 3 || an(i21) || an(i11) || an(i 1) || an(i11) ] in BNCi 1 for the NC
NC (i 1) [0 || 0 || 0 || 0 || ti(i 1) || ti(i 11) || || tn(i31) || tn(i21) || tn(i11) ||
|| tn( i 1) || tn( i11) ] ; ti( i 1) ai( i 1) ; ti(i 1) 0, mi 1 .
Performing a subtraction operation A(i ) A( i 1) NC ( i 1) [0 || 0 || 0
For
... || 0 || ai(i 1) || ai(i 11) || ... || an(i31) || an( i21) || an( i11) || an( i 1) || an( i11) ] [0 || 0 || 0 || 0 ||
value
A( i ) || t(i 1)
i
(i 1)
|| t
i 1 || || t ( i 1)
n 3 || t ( i 1)
n2 || t ( i 1)
n 1 || t ( i 1)
n || t ( i 1)
n 1 ] 0 || 0 0 ... 0
( i 1) ( i 1) ( i 1) ( i 1)
|| [a i t i ]mod mi [a i 1 t i 1 ]mod mi 1 || ... [an( i31) tn( i31) ]mod mn 3
|| [an(i21) tn( i21) ]mod mn 2 [an( i11) tn( i11) ]mod mn 1 || [ an( i 1) tn( i 1) ]mod mn
|| [an(i11) tn(i11) ]mod mn1 [0 || 0 || 0 || 0 0 ai(i )1 || an(i)3 || an(i)2 || an(i)1
|| an( i ) || an( i)1 ] .
Appeal by the value of the remainder an( n1 2) of a number
( n 2)
A [0 || 0 || 0 ||
2n 3 ... || 0 || 0 || 0 || 0 0 || an( n1 2) || an( n 2) || an( n1 2) ] in for the NC
BNCn 2
( n 2) ( n 2) ( n 2)
NC [0 || 0 || 0 || || 0 || 0 || 0 || || 0 || 0 || t n 1 || t n ||
|| tn( n1 2) ] ; tn( n1 2) an( n1 2) ; tn( n1 2) 0, mn 1 1 .
Performing a subtraction operation A( n 1) A( n 2) NC ( n 2) [0 || 0 || 0 |
... || 0 || 0 || 0 || || 0 || 0 || an( n1 2) || an( n 2) || an( n1 2) ] [0 || 0 || 0 || || 0 || 0 || 0 ||
2n 2 ... || 0 || 0 || t ( n 2)
n 1 || t ( n 2)
n || t ( n 2)
n 1 ] 0 || 0 0 ... || 0 0 0 || ... 0 0
( n 2) ( n 2)
|| [a n 1 t n 1 ]mod mn 1 || [an( n 2) tn( n 2) ]mod mn
|| [an( n1 2) tn( n1 2) ]mod mn1 [0 || 0 || 0 || || 0 || 0 || 0 || 0 || 0 || 0 an( n1) ||
|| an( n11) ] .
Appeal by the value of the remainder an( n 1) of a number
( n 1) ( n 1) ( n 1)
2n 1 A [0 || 0 || 0 || ... || 0 || 0 || 0 || || 0 || 0 || 0 || a n || a n 1 ] in BNCn 1
( n 1)
for the NC NC [0 || 0 || 0 || || 0 || 0 || 0 || || 0 || 0 || 0 || || tn( n 1) || tn( n11) ] ;
tn( n 1) an( n 1) ; tn( n 1) 0, mn 1 .
Getting a nullified number A( Z ) A( n) A( n1) NC(n1)
[0 || 0 || 0 || || 0 || 0 || 0 || ... || 0 || 0 || 0 || an || an( n11) ] [0 || 0 || 0 || || 0 || 0 || 0 ||
( n 1)
2n ... || 0 || 0 || 0 || tn( n 1) || tn( n11) ] 0 || 0 0 ... || 0 0 || 0 || ... 0 0 0
|| [an( n 1) tn( n 1) ]mod mn [an( n11) tn( n11) ]mod mn 1 [0 || 0 || 0 || || 0 || 0 || 0 ||
... 0 || 0 || 0 0 a (n)
n 1 ] , где a (n)
n 1 n 1 .
TZ 1 2 n add
Fig. 1. Procedure of SN of numbers in the SRC
Denoting the sampling time of the NC from the corresponding nullification block
(NB) of the CS functioning in the SRC as t1, and the time of subtracting from the
number А(i-1) the constant NC(i-1), i.e. performing the operation А(i) = А(i-1) NC(i-1) – as
t2, we get the total time ТZ1 of the nullification procedure for the first Z1 method:
ТZ1 = n (t1 + t2). (6)
3 Parallel subtraction method
Consider the following method (Z3) of operational control of data in the SRC (par-
allel nullification method (PNM)). The essence of the proposed control method is that
the nullification procedure is carried out parallel in time for two reasons. For n -even
numbers, we have аі(i–1), an( ii1)1 ( i 1, n / 2 ), namely a1(0), an`(0); a2(1), an–1(1); a3(2), an–
(2)
2 ;… an( n/ 2/ 2) , an( n/ 2) /2)1 (see Fig. 2). For n -odd numbers, we have a1(0), an(0); a2(1), an–1(1);
a3(2), an( 2) 2 ; … a(((nn1)/1)/22 1) (see Fig. 3). In this case, for an arbitrary value i of NC for
i zeroes
the corresponding number, have the following form: A(i ) [0||0|| ||0|| ai(i )1 || ai(i )2 ||...
i zeroes
...|| an( i)i 1 || an( i)i ||0||||0||0|| an( i)1 ] ,
NC (i ) [0||0|| ||0|| ti(i )1 || ti(i )2 || ||tn(i)i 1 ||tn(i)i ||0|| ||0||0|| || tn( i)1 ] ; ti(i )1 0, mi 1 ,
tn(i)i 0, mn i ; ti(i )1 ai(i )1 , tn(i)i an(i)i .
a1(0) a2(1) a3( 2) . . . ai( i 1) ai(i 11) . . . an(2)1
an(1) an(0)1
an( n/ 2/ 2 1) an( n/ 2/ 211)
a3( 2) an( 2) 2
a2(1) an(1)1
a1(0) i 1, n / 2
an(0)
Fig. 2. Sampling scheme for nullification constants for PNM method (n –even number)
For an arbitrary value i we have that:
A(i 1) A(i ) NC(i )
[0 || 0 || || 0 || a (i )
i 1 || a(i )
i2 || ai(i )3 || ... || an(i)i 2 || an(i)i 1 || an(i)i || 0 ... 0 || an(i)1 ]
[0 || 0 || ... || 0 || ti(i )1 || ti(i )2 || ti(i )3 || || tn( i)i 2 || tn( i)i 1 || tn( i)1 || 0 ... 0 || tn( i)1 ]
0 || 0 ... 0 [ai(i )1 ti(i )1 ]mod mi 1 ||[ai(i )2 ti(i )2 ]mod mi 2 || [ai(i )3 ti(i )3 ]mod mi 3 || ...
... [an( i)i 2 tn(i)i 2 ]mod mn i 2 [an(i)i 1 t n(i)i 1 ]mod mn i 1
|| [an(i)i tn(i)i ]mod mni 0 ... 0 [an(i)1 tn(i)1 ]mod mn1
[0 || 0 || ... 0 0 || ai(i 21) || ai(i 31) || ... an( ii1) 2 || an( ii1)1 || 0 0 ... 0 0 || an( i11) ] .
a1(0) a2(1) a3( 2) ... ai( i 1) ai(i )1 ai(i 21) ... an( 2) 2 an(1)1
an(0) an 1
a((nn1)/
1)/ 2 1
2
a((nn1)/
1)/ 2 1
2 a((nn1)/
1)/ 2 1
2 1
a3( 2) an( 2) 2
a (1)
2
an(1)1
i 1,(n 1) / 2
a1(0) a1(0)
Fig. 3. Sampling scheme for nullification constants for PNM method (n – odd number)
The algorithm for performing the PNM procedure is presented in Table. 2. Before
getting the value n 1 an( n/12) for n - even number, we have that:
n / 2 1zeroes n / 2 1 zeroes
( n / 2 1) ( n / 2 1) ( n / 2 1)
A [0 || 0 || || 0 || a n/ 2 || a n / 2 1 || 0 || || 0 || 0 || an( n/12 1) ] .
n / 2 1zeroes n / 2 1zeroes
( n / 2 1) ( n / 2 1) ( n / 2 1)
NC [0 || 0 || || 0 | | t n/ 2 || t n / 2 1 || 0 || || 0 || 0 || tn( n/12 1) ] ,
tn( n/2/21) 0, mn /2 , tn( n/ 2/ 211) 0, mn / 21 , tn( n/ 2/ 2 1) an( n/ 2/ 21) , tn( n/ 2/ 211) an( n/ 2/ 211) .
A( z ) A( n / 2) A( n / 2 1) NC ( n / 2 1) 0 || 0 || || 0 || an( n/ 2/ 2 1) tn( n/ 2/ 2 1) mod mn / 2 ||
|| an( n/ 2/ 211) tn( n/ 2/ 211) mod mn / 2 1 ||0 || 0 || || 0 || 0 || an( n/12 1) tn( n/12 1) mod mn 1
0 || 0 || ||0||0|| an( n/12) , where n 1 an 1 .
( n / 2)
|| 0 || 0||0||
Before getting the value n 1 an( n/12) for n -odd number, we have that
n 1 n 1
1 zeroes 1 zeroes
2 2
A(( n 1)/ 2 1) [0 || 0 || || 0 || a(((nn1)/1)/22 1) || 0 || || 0 || an((n11)/ 2 1) ] .
Table 2. PNM algorithm
Operation
Contents of operation
number
Appeal by the value of the remainders a1(0) and an(0) of a number A(0) in
1 BNC0 for the NC(0) .
2 Performing a subtraction operation A
(1)
A(0) NC(0) .
Appeal by the value of the remainders a2(1) and an(1)1 of a number A(1) in BNC1
3 for the NC .
(1)
4 Performing a subtraction operation A
(2)
A(1) NC(1) .
Appeal by the value of the remainders a2( 2) and an( 2) 2 of a number A( 2) in
5 BNC2 for the NC(2) .
6 Performing a subtraction operation A
(3)
A(2) NC(2) .
… …
i Performing a subtraction operation A
(i )
A(i 1) NC(i 1) .
Appeal by the value of the remainders ai(i )1 and an( i)i of a number A( i ) in
i 1 BNCi for the NC (i ) .
(i 1)
i2 Performing a subtraction operation A A(i ) NC (i ) .
… …
Appeal by the value of the remainders an( n/ 2/ 21 2) and an( n/ 2/ 222) of a number
n3 A( n / 2 2) in BCN n / 2 2 for the NC
( n /2 2)
.
( n /21)
n2 Performing a subtraction operation A A( n/22) NC( n/22) .
Appeal by the value of the remainders an( n/ 2/ 2 1) and an( n/ 2/ 211) of a number A( n / 2 1)
n 1 in BNCn / 2 1 for the NC
( n /21)
.
Performing a subtraction operation A
( n /2)
A(n/21) NC( n/21) .
Getting nullified number A( Z )
n A( Z ) A( n / 2) [0 || 0 || ... || 0 || ... || 0 || 0 || n 1 an( n/12) ] .
NC (( n1)/21) [0 || 0 || || 0 || t(((nn1)/2
1)/2 1)
|| 0 || || 0 || tn((n11)/2 1) ] ,
t(((nn1)/1)/22 1) 0, m( n 1)/ 2 ; t(((nn1)/2
1)/2 1)
a(((nn1)/2
1)/2 1)
.
A( Z ) A(n1)/2 A((n1)/21) NC((n1)/21)
0 || 0 || || 0 || a(((nn1)/1)/22 1) t(((nn1)/1)/22 1) mod m( n 1)/ 2 ||0||...0 || 0 || || 0 ||0||
|| an((n11)/ 2 1) tn((n11)/ 2 1) mod mn 1 0 || 0 || || 0 || ||0||0|| an( n11)/ 2 , where
n 1 an( n11)/ 2 .
PNM method in the SRC presented in Fig. 4.
Opera-
tion
Contents of operation
number
(cycle)
Appeal by the value of the remainders a1(0) and an(0) of a number
A A(0) [a1(0) ||
|| a2(0) || a3(0) || || ai(0) (0)
1 || ai || ai(0) (0) (0) (0) (0) (0)
1 || ... || an 3 || an 2 || an 1 || an || an 1 ] in
1
BNC0 for the NC (0)
[t (0)
1 || t (0)
2 || t (0)
3 || || t (0)
i 1 || t (0)
i || t (0)
i 1 ||
... || t (0)
n 3 || t (0)
n2 || t (0)
n 1 || t (0)
n || t (0)
n 1 ]; t (0)
1 a , (0)
1 t (0)
n a ; (0)
n t1(0) 0, m1 1 ,
tn(0) 0, mn 1
Performing a subtraction operation A(1) A(0) NC (0) [a1(0) || a2(0) ||
|| a3(0) || || ai(0) (0)
1 || ai || ai(0)
1 || || an(0)3 || an(0) 2 || an(0)1 || an(0) || an(0)1 ] [t1(0) || t2(0) ||
|| t3(0) || || ti(0) (0)
1 || ti || ti(0)
1 || || tn(0)3 || tn(0) 2 || tn(0)1 || tn(0) || tn(0)1 ]
[a1(0) t1(0) ]mod m1 || [a2(0) t2(0) ]mod m2 [a3(0) t3(0) ]mod m3 || ...
2
1 ti 1 ]mod mi 1 || [ ai
... || [ai(0) ti(0) ]mod mi [ai(0)
1 ti 1 ]mod mi 1 || ...
(0) (0) (0)
... [an(0)3 tn(0)3 ]mod mn 3 || [an(0) 2 tn(0) 2 ]mod mn 2 [an(0)1 tn(0)1 ]mod mn 1 || ...
... || [an(0) tn(0) ]mod mn || [an(0)1 tn(0)1 ]mod mn 1 [0 || a2(1) || a3(1) ||
... || ai(1)1 || ai(1) || ai(1)1 || || an(1)3 || an(1) 2 || an(1)1 || 0 || an(1)1 ] .
Appeal by the value of the remainders a2(1) and an(1)1 of a number
A(1) [0 || a2(1) ||
|| a3(1) || || ai(1)1 || ai(1) || ai(1)1 || || an(1)3 || an(1) 2 || an(1)1 || 0 || an(1)1 ] in BNC1 for
3
the NC (1) [0 t2(1) || t3(1) || || ti(1) (1)
1 || ti || ti(1)
1 || || tn(1)3 ||
|| tn(1) 2 || tn(1)1 || 0 || tn(1)1 ] ; t2(1) a2(1) , tn(1)1 an(1)1 ; t2(1) 0, m2 1 ,
tn(1)1 0, mn1 1 .
Performing a subtraction operation A(2) A(1) NC (1) [0 || a2(1) || a3(1) ||
... || ai(1)1 || ai(1) || ai(1)1 || || an(1)3 || an(1) 2 || an(1)1 || 0 || an(1)1 ] [0 || t2(1) || t3(1) ||
... || ti(1) (1) (1)
1 || ti || ti 1 || || tn(1)3 || tn(1)2 || tn(1)1 || 0 || tn(1)1 ] 0 || [a2(1) t2(1) ]mod m2
4 || [a3(1) t3(1) ]mod m3 || [a4(1) t4(1) ]mod m4 || ... || [ai(1)1 ti(1)
1 ]mod mi 1 ||
|| [ai(1) ti(1) ]mod mi [ai(1)1 ti(1)
1 ]mod mi 1 || ... [ an 3 t n 3 ]mod mn 3 ||
(1) (1)
|| [an(1)2 tn(1)2 ]mod mn2 [an(1)1 tn(1)1 ]mod mn1 || 0 || [an(1)1 tn(1)1 ]mod mn1
[0 || 0 || a3(2) || a4(2) || || ai(2) (2)
1 || ai || ai(2)
1 || || an(2)3 || an(2) 2 || 0 0 || an(2)1 ] .
Appeal by the value of the remainders a3( 2) and an( 2) 2 of a number
A(2) [0 || 0 ||
5 || a3(2) || || ai(2) (2)
1 || ai || ai(2)
1 || || an(2)3 || an(2) 2 || 0 || 0 || an(2)1 ] in BNC2 for the
NC (2) [0 || 0 || t3(2) || || ti(2) (2)
1 || ti || ti(2) (2) (2)
1 || ... || t n 3 || t n 2 ||
|| 0 || 0 || tn(2)1 ] , t3(2) a3(2) , tn(2) 2 an(2) 2 ; t3(2) 0, m3 1 , tn(2)2 0, mn 2 1 .
Performing a subtraction operation A(3) A(2) NC (2) [0 || 0 || a3(2) ||
... || ai(2) 1 || ... || an 3 || an 2 || 0 || 0 || an 1 ] [0 || 0 || t3 || ... || ti 1 || ti
(2)
1 || ai || ai(2) (2) (2) (2) (2) (2) (2)
||
|| ti(2)
1 || || tn(2)3 || tn(2)2 || 0 || 0 || tn(2)1 ] 0 || 0 [a3(2) t3(2) ]mod m3 ||
|| [a4(2) t4(2) ]mod m4 || ... || [ai(2)
1 ti 1 ]mod mi 1 || [ ai
(2) (2)
ti(2) ]mod mi
6
1 ti 1 ]mod mi 1 || ... || [ an 3 tn 3 ]mod mn 3 || [ an 2 tn 2 ]mod mn 2 ||
|| [ai(2) (2) (2) (2) (2) (2)
0 0 [an(2)1 tn(2)1 ]mod mn1 [0 || 0 || 0 || a4(3) || a5(2) || || ai(3) (3)
1 || ai || ai(3)
1 ||
... || an(3) 4 || an(3)3 || 0 0 0 || an(3)1 ] .
Appeal by the value of the remainders ai( i 1) and an( ii1)1 of a number
A( i 1) [0 || 0 || ...
For ... || 0 || ai(i 1) || ai(i 11) || ai(i 21) || ... || an( ii1)3 || an( ii1) || an( ii1)1 || 0 || 0 ... 0 || an( i11) ] in
value ( i 1)
BNCi 1 for the NC [0 || 0 ... 0 ||
A( i )
( i 1) ( i 1) ( i 1)
|| ti || t
i 1 || t
i2 || || tn(ii1)1 || tn(ii1) || tn(ii1)1 0 || 0 || ... 0 tn(i11) ] ; ti( i 1) ai( i 1)
, tn(ii1)1 an( ii1)1 ; ti(i 1) 0, mi 1 , tn(ii1)1 0, mn i 1 1 .
Performing a subtraction operation A(i ) A( i 1) NC ( i 1) [0 || 0 || 0 || ...
... || 0 || ai(i 1) || ai(i 11) || || 0 || 0 || an( i11) ] [0 || 0 || 0 || || 0 || ti( i 1) || ti(i 11) || ... || 0 ||
|| 0 || tn(i11) ] 0 || 0 ... 0 [ai(i 1) ti(i 1) ]mod mi || [ai(i 11) ti(i 11) ]mod mi 1 ||
[ai(i 21) ti(i 21) ]mod mi 2 || ... [an(ii1)1 tn(ii1)1 ]mod mn i 1
|| [an( ii1) tn( ii1) ]mod mn i || [an( ii1)1 tn( ii1)1 ]mod mn i 1 0 || 0 ... 0
|| [an(i11) tn(i11) ]mod mn1 [0 || 0 || ... 0 0 || ai(i )1 || ai(i )2 || ai(i )3 || ...
... || an(i)i 1 || an(i)i || 0 0 ... 0 || an(i)1 ] .
Appeal by the value of the remainders ai(i )1 and an( i)i of a number
A( i ) [0 || 0 || 0 || ...
... || 0 || 0 || ai(i )1 || || an(i)i 1 || an(i)i || 0 || 0 || an(i)1 ] in BNCi for the
NC (i )
[0 || 0 || 0 || || 0 || 0 || t (i )
i 1 || || t (i )
n i 1 || t (i )
n i || 0 || 0 || t (i )
n 1 ]; t (i )
i 1 ai(i )1 ,
tn(i)i an(i)i ; ti(i )1 0, mi 1 1 , tn(i)i 0, mn i 1 .
For Performing a subtraction operation A(i 1) A(i ) NC (i ) [0 || 0 ||
value ... || 0 || ai(i )1 || ai(i )2 || ai(i )3 || ... || an( i)i 2 || an( i)i 1 || an( i)i || 0 || ... || 0 || an( i)1 ]
A(i 1)
[0 || 0 || || 0 || ti(i )1 || ti(i )2 || ti(i )3 || || tn(i)i 2 || tn(i)i 1 || tn( i)1 || 0 ... 0 || tn( i)1 ]
0 || 0 ... 0 || [ai(i )1 ti(i )1 ]mod mi 1 || [ai(i )2 ti(i )2 ]mod mi 2 ||
|| [ai(i )3 ti(i )3 ]mod mi 3 || ... [an( i)i 2 tn( i)i 2 ]mod mn i 2
|| [an(i)i 1 tn(i)i 1 ]mod mn i 1 || [an(i)i tn(i)i ]mod mn i 0 ... 0
|| [an(i)1 tn(i)1 ]mod mn1 [0 || 0 || ... 0 0 || ai(i 21) || ai(i 31) || ... an(ii1)2 || an(ii1)1 ||
|| 0 0 ... 0 0 || an(i11) ] .
Further for n even n odd numbers we get: for n even number. Appeal by
the value of the remainders an( n/ 2/ 2 1) and an( n/ 2/ 211) of a number
A( n / 2 1) [0 || 0 || || 0 || an( n/ 2/ 2 1) || an( n/ 2/ 211) || 0 || 0 || 0 || an( n/12 1) ] in BNCn / 21
for the NC ( n / 2 1) [0 || 0 || || 0 || tn( n/ 2/ 2 1) ||
|| tn( n/ 2/ 211) || 0 0 0 || tn( n/12 1) ] ; tn( n/ 2/ 2 1) an( n/ 2/ 2 1) , tn( n/ 2/ 211) an( n/ 2/ 211) ;
n 1
tn( n/ 2/ 21) 0, mn / 2 1 , tn( n/ 2/ 211) 0, mn / 21 1 .
For n odd number.
Appeal by the value of the remainder a(((nn1)/1)/22 1) of a number
(( n 1)/ 2 1)
A [0 || 0 ||
(( n 1)/ 2 1)
... || 0 || a ( n 1)/ 2 || 0 || 0 || an((n11)/ 21) ] in BNC( n 1)/ 2 1 for the
NC (( n1)/ 21) [0 || 0 || || 0 || t(((nn1)/1)/221) || 0 0 || tn((n11)/ 21) ] ,
(( n 1)/ 2 1)
t(((nn1)/1)/221) a(((nn1)/1)/221) ; t( n 1)/ 2 0, m( n 1)/ 2 1 .
For n even and n odd numbers, we obtain the following values of the
number A( Z ) to be nullified.
For n even number.
Getting a nullified A( Z ) number: A( Z ) A( n / 2) A( n / 2 1)
NC ( n / 2 1) [0 || 0 || || 0 || an( n/ 2/ 2 1) || an( n/ 2/ 211) || 0 || 0 || 0 || an( n/12 1) ]
[0 || 0 || || 0 || tn( n/ 2/ 21) || tn( n/ 2/ 211) || 0 || 0 || 0 || tn( n/121) ] 0 || 0 ... 0
|| [an( n/ 2/ 2 1) tn( n/ 2/ 2 1) ]mod mn / 2 || [an( n/ 2/ 211) tn( n/ 2/ 211) ]mod mn / 2 1 || 0 ... 0 0
|| [an( n/2
1
1)
tn( n/2
1
1)
]mod mn1 0 || 0 || ... 0 0 || 0 || ... 0 0 an( n/2)
1 , where
n n 1 an( n/12) .
For n odd number.
Getting a nullified A( Z ) number: A( Z ) A( n 1/ 2) A(( n 1/ 21)
NC (( n 1)/ 21) [0 || 0 || || 0 || a(((nn1)/1)/221) || 0 || 0 || an((n11)/ 2 1) ] [0 || 0 ||
... || 0 || t(((nn1)/1)/221) || 0 || 0 || tn((n11)/ 2 1) ] 0 || 0 ... 0 ||
|| [a(((nn1)/1)/221) t(((nn1)/1)/221) ]mod m( n 1)/ 2 || 0 ... 0
|| [an((n11)/21) tn((n11)/21) ]mod mn1 [0 || 0 || ... 0 ... 0 0 || an((n11)/2) ] ,
where n 1 an((n11)/ 2) .
TZ 3 n
Fig. 4. Procedure of SN of numbers in the SRC
The time ТZ3 for performing the zeroing procedure for the first (Z3) method of the
PNM is defined as:
ТZ3=nadd. (7)
When implementing the nullification procedure for the second (Z3) method in the
block of nullification constants (NB) of the calculator in the SRC, it is necessary to
n
2
have K Z 3 (mi mn i 1 1) nullification constants. In this case, the number of NZ3
i 1
double digits of the NB nullification constants is determined by the expression
n
2
K Z 3 (mi mn i 1 1) (n 2i 1) .
i 1
4 Conclusions
The article presents the nullification of numbers in the system of residual classes
(SRC). This method is widely used in the non-positional number system in the SRC
with the need to determine positional characteristics. Two methods of nullification are
presented in the article: the method of successive subtractions and the method of par-
allel subtractions. Based on these methods, algorithms are developed for their imple-
mentation. The essence of the method of successive subtractions is that the nullifica-
tion procedure is carried out consequently from the junior foundation to the oldest.
The essence of the parallel subtraction method is that the nullification procedure is
carried out parallel in time for two reasons.
It is advisable to use these methods in the implementation of operations of compar-
ing numbers in the SRC, and in monitoring data presented in the SRC [9-13]. The
estimation of these methods by the number of equipment and by the time of imple-
mentation of the nullification procedure is made. In terms of the efficiency of the
nullification procedure, a parallel subtraction method was proposed.
The obtained research results can be useful for various methods of increasing the re-
liability and fault tolerance of computer systems [14-18].
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