Methods of data control in the residue classes are considered in the article. The main advantage of non-positional notation in the residue classes lays in the possibility of an organization of the process of quick implementation of modular arithmetic operations of addition, subtraction and multiplication. The base of the method is the procedure of generating and using the positional indication of non-positional code. That allows increasing efficiency of the procedure of data control granted by the residue classes.
It is well known, that the main advantage of non-positional notation in the residue
classes (NRC) is laid in the possibility of quick process of organization of modular
arithmetic operations of addition, subtraction and multiplication [
However, in computer systems for common purpose it is needed to perform socalled non-modular (position) operations except the above listed arithmetic operations. The following operations belong to these: Arithmetic and algebraic congruence of numbers and its absolute values; Sign of number definition; Definition of existing of overflow of bit grid of computer system (CS); Rounding of value of result of operation; Computing the absolute value; Division and multiplication of fraction Translation of data from code of NRC to positional notation and back; Expanding of original NRC (it is an informational process in which familiar remainders {ai } , that correspond to basis {mi } , define value of remainders with the same code structure by other additional basis); Control, diagnostic and correction mistakes of NRC data, etc.
Generally, all positional operations come down to the procedure of definition of an
index of j numerical jmi , j 1 mi interval of entering (detecting) of number
А а1,..., аi–1, аi , аi1,..., an , an1 . It is appropriate to use so-called positional
indications of non-positional code (PINC) to define an index of j numerical interval of
detecting A. The following indications are the most frequently used in NRC (within
existing PINCs) [
There are disadvantages of the above-mentioned PINC. At first, it is technical and
time complexity of PINC generating (developing) for given code structure
А а1,..., аi–1, аi , аi1,..., an , an1 . At second, it is no mean time of implementation
having applied the existing positional indications, non-modular operations in NRC,
specifically, operations of control, diagnostic of correction of data mistakes [
It becomes obvious that the research of developing methods of generating new PINC in NRC is important; because of them, we can use operatively non-modular operations. It should be pointed out, population (sequence) of defined modular and non-modular operations, which are implemented by PINC, can realize any nonmodular operation.
Beforehand we will consider the general requirements to PINC, on base of which in article will be developed method of increasing of operational efficiency of data control: To define exactly correctness and incorrectness of number A in NRC (to define fact of detecting/not detecting of number A in an informational numerical [0, M ] Simplicity
n interval, where M mi ) by used (chosen, developed, generated) PINC; i1 of generating PINC for given code structure of data А а1,..., аi–1, аi , аi1,..., an , an1 ; Simplicity of using of generated indication for data control in NRC (in general case for implementing positional operations); Indication has to have clear and understandable physical meaning; Indication must be analytically described by simple mathematical relator; It is possible to technically implement process of data control in NRC by using
Using the chosen indication of non-positional code has to provide the given fidelity of data control in NRC; PINC using has to provide the possibility of exception from the control procedure, diagnostic and correction of mistakes in NRC the most difficult positional operations.
It is reasonable to develop and research the method of data control in NRC on the base of PINC using, resulting from the above-mentioned information. 2
Method of generating of positional indication of nonpositional code structure of data in NRC
The indication nA forms proceeding from representation of initial code structure А а1,..., аi–1, аi , аi1,..., an , an1 , which is represented as NRC with basis {mi} , i 1, n 1 , in the way of creating of so-called uniserial code (UC).
K N(nA ) Z N(A)1 Z N(A)1 ... Z1( A) Z ( A) 0 represents sequence of binary bits, which contains ones and a zero, index of which is at nA place (the count is performed from right to left, from digit bit Z ( A) to digit bit 0
nA jmi , j 1 mi
defines an index of j numerical interval of detecting of number А а1,..., аi–1, аi , аi1,..., an , an1 .
Mathematically PINC nA represents a positive integer, which points out the location of zero binary bit in the record of UC K N(nA ) Z ( A) 0 . nA
KH ( A) a1' ,..., ai'1, ai , ai'1,..., an'1
mi defines in the block of constants of nuvelization (BCN). It is performed by the definition of remainder ai of number А а1,..., аi–1, аi , аi1,..., an , an1 for chosen integer mi of NRC.
Further, using the chosen constant of nuvelization KH ( A) , we displace number A mi on the left edge of the interval jmi , j 1 mi by implementing operation Ami A KH m(Ai) a1 , a2 , ..., ai1 , ai , ai1 , ..., an1 a1' , a2' , ..., ai'1 , ai , ai'1 ,..., an'1 a1(1) , a2(1) , ..., ai(1)1, 0, ai(1)1 , ..., an(1)1 .
A (a1,a2,...,ai1,ai,ai1,...,an,an1) , GCD (mi,mj ) 1,i j . 2. 3.
Choice of basis mi {mi}, ( j 1,n 1) , by which defines an index of j numerical interval jmi, j 1mi of detecting number А.
KHm(A) (a1' ,a2',...,ai'1,ai,ai'1,...,an',an'1)
i by value of remainder ai (by module mi ) of number A.
Ami A KHm(Ai) a1,a2,...,ai1,ai,ai1,...,an,an1 a1',a2',...,ai'1,ai,ai'1,...,an,an1 a(1),a2(1),...,ai(1)1,0,ai(1)1,...,an(1),an(1)1. 1
n1 K1
Ki KN(nA) ZN(A)1 ZN(A)2 ... Z0(A), KN(niA) ZN(Ai)1 ZN(Ai)2 ... Z0(A).
N mK , Ni ]M / mi[ , M mi . Am KA mi ZK(AA) .
i n i1 m 0mi Z(A), A Ami 1...mi Z(0A), i
1 Am (N 2)mi ZN(A)2,
i Am (N 1)mi Z(A)
i N1. Am 0mi Z(A), Ami 1...mi Z(0A), i
1 Ammi ((NNii 12))mmii ZZ(N(AAi))2.,
i A
Ni1
Defining PINC nA of number A, which means numerical value of nA for 6 which Z(A) Z(A) 0 , which means Am nA mi 0 .
KA nA i Herewith Zl(A) 1, (Am l mi 0; l nA) .
i PINC nA has to define the fact of entering or not entering initial number A in the interval [0, M ) . We have to implement operation to define the fact of detecting number in numerical informational interval [0, M).
The operation (1) is carried out simultaneously using the population of N constants Ami K A mi Z K(AA) K A mi K A 0, N 1 , (1) (2) (3) n1 where N mK :
K 1
K i which means value
In this case, UC represents in the view mi 0 mi Z0( A) , A AAmi 1 mi Z1( A) , mi .2..mi Z2( A) , Ami (N 2) mi Z N(A)2 , Ami (N 1) mi Z N(A)1.
K N(nA ) Z N(A)1 Z N(A)2 ... Z1( A) Z ( A) 0 The only value nA from (1) exists in total (2) analytical rations. For nA there is Z ( A) Z ( A) 0 (K A nA ) ,
K A nA
Ami nA mi 0 .
Zl( A) 1 ( Ami l mi 0; l nA ) .
n1 N mK .
K 1
K i
It should be pointed out, that there is no necessity to have the whole sequence of values Z ( A) from (3) to define only the fact of the garbling of number ( A M ) . It is
K A enough to have UC K (nA ) with length only Ni ]M / mi[
Ni value ]M / mi [ describes the lesser integer of number M / mi ).
In this case values of variables of numerical intervals jmi , j 1 mi , which are out of informational interval 0, M , make no matter for establishing the fact of correctness control of number A. of binary bits (where 3
Method of operational data control in NRC on base of using positional indication of non-positional code The procedure of generating of positional indication of non-positional code (fig. 1) laid in the base of method of operational data control in residue classes. In that way, the essence of the method of data control in residue classes lies in the following. For the controlled code structure А а1,..., аi–1, аi , аi1,..., an , an1 , which is represented in residue classes, developed (defined) PINC nA by generating UC
K N(niA ) Z N(Ai)1 Z N(Ai)2 ... Z1( A) Z0( A) in view of sequence of Ni binary bits. Choosing of basis mi of NRC is performed by special-purpose in accordance with defined measures. The constant of nuvelization KH ( A) a1', a2' ,..., ai ,..., an' , an'1
mi is selected depending on a result of the value of remainder ai of number A. Further implementation operation is carried out:
Ami A KH m(Ai) a1, a2 ,..., ai ,..., an , an1 a1' , a2' ,..., ai ,..., an' , an'1 a(1) , a(1)
1 2 ,..., 0,..., an(1) , an(1)1 .
Using Ni constants K A mi K A 0, Ni 1 simultaneously the operations of subtractions Ami K A mi are carried out, in the result of which appears the values of binary bits Z ( A) , so the UC K (nA ) forms.
K A Ni
If nA Ni , number A is considered as a wrong. In the opposite case ( nA Ni ) number A is correct. In common view, method of data control is represented on Fig. 2.
Ami nA mi 0 , Zn( AA) 0 ; Zl( A) 1, Ami l mi 1; l nA . Data control A a1, a2 ,..., ai1, ai , ai1,..., an , an1 in NRC.
we define that nA 9
K (nA) K3(99) {11...110111111111} .
Ni
Amn1 nA mn1 0 , ( Amn1 nA mn1 99 9 11 0 ), meaning Z (A) Z9(A) .
nA
Check: А 100 M 420 (number A is right). Example 2. Perform control of data A (00,10, 000, 010,1010) .
KH m(An)1 (01,10, 000, 011,1010) is selected by value of a5 1010 in BCN (Table 1).
A KH ( A) (10, 00, 000,110, 0000) .
mn1
Amn1 nA mn1 440 44 11 0 then UC has view K (nA ) K3(940) {11...11...11} and nA 40 .
Ni
Check: А 450 M 420 (number A is wrong). Example 3. Perform control of data А 01,11, 010, 000,1001. .
is selected by value of a5 1001 in BCN (Table 1). We deduce that KH ( A) (00, 01,100, 010,1001)
mn1 Amn1 A KH ( A) (01,10, 011,101, 0000) .
mn1
then UC has view and nA 38 .
Amn1 nA mn1 418 38 11 0
Because nA 38 Ni 39 of it means that number A is right (ungarbled). Though the check А 427 M 420 shows us that number A is wrong (Fig. 3).
The represented method can be used for improving promising computer systems
and their components and in the other practically important applications [
Conclusions Thus, the method of data control in the system of residue classes is presented in the article. The procedure of forming and using of position indication of non-positional code is the base for a method of operational control of data in a residue class. Use of PINC allows to increase efficiency of the procedure of data control provided to NRC. It should be pointed out, that any non-modular operation can be implemented by set (sequence) of defined modular and non-modular operations, which are implemented by PINC. Using PINC in the method provides the possibility of exception of the most complicated positional operations from the procedure of control, diagnostic and correction of mistakes in NRC.