The data errors control in the modular number system based on the nullification procedure

. A method for error control in the modular number system (MNS) based on the use of the zeroing procedure is proposed. Error control in the MNS is a non-positional operation and requires the development of special methods, designed to increase the efficiency of this procedure. This method is designed to verify the correct implementation of the computing process of computer systems and components. It is assumed that the error in one module remainder does not affect the residual values corresponding to other modules (bases) of the MNS. The essence of the proposed method is that, when performing the procedure of zeroing in the MNS, the operation of determining is combined in time, in accordance with the digits ( 1 ) ii a  and ( i 1 ) 1 n i a  of the number ( i 1 ) A  , the zeroing constant ( i ) Z C and the calculation operation for the values of ( i 1 ) i a  and ( i 1 ) 1 n i a  of the following digits ( ) 1 ii a  and ( )- i n i a of the number (i) A . This makes it possible to increase the efficiency of information control,


Abstract.
A method for error control in the modular number system (MNS) based on the use of the zeroing procedure is proposed. Error control in the MNS is a non-positional operation and requires the development of special methods, designed to increase the efficiency of this procedure. This method is designed to verify the correct implementation of the computing process of computer systems and components. It is assumed that the error in one module remainder does not affect the residual values corresponding to other modules (bases) of the MNS. The essence of the proposed method is that, when performing the procedure of zeroing in the MNS, the operation of determining is combined in time, in accordance with the digits ( 1)  A . This

Introduction
The main direction of modern science and technology is the development and use of new advanced information technologies based on the extensive use of computer systems and components (CSC). Information technologies are increasingly invading our lives, penetrating all processes (social, economic, political). Scales and complexity of the tasks solved by modern computer systems impose qualitatively new requirements to their main characteristics: productivity, reliability and efficiency of systems that causes need of improvement existing and creations of new means of information processing. In modern computer systems, an improvement in one group of quality indicators, for example, an increase in productivity, leads to a deterioration in othersa complication of structure, an increase in cost, a decrease in reliability, etc. [1].
In connection with the constant complication of scientific and technical problems of processing integer data, the trend of development of CSC is aimed at increasing the speed (productivity) and reliability of the implementation of integer arithmetic operations [1][2]. The results of recent years in the field of information technology by various groups of the scientists and engineers of methods for increasing the productivity, reliability, survivability and also reliability of calculations of computer systems have shown that it is practically impossible to achieve this within the limits of the positional number systems (PNS). This is due to the main disadvantage of modern CSC, functioning in the PNS: the presence of inter-digit relations between the processed numbers. These relations negatively affect the architecture of the CSC and the methods of implementing arithmetic operations, complicate the equipment, they limit the speed and reliability of performing arithmetic operations. In this regard, in the PNS, the increase in the performance of the CSC is achieved by increasing the clock frequency, as well as through the use of methods and tools for parallel data processing, and also by use of different types of reservation [3].
Application of the basic methods of increasing the productivity of the CSC, based on the parallelization of computations, by using some properties of solvable tasks and algorithms cannot increase the productivity of CSC in each and every case. The scope of their application is limited to a class of tasks to be solved. In addition, the process of artificial dismemberment of the algorithm itself, the determination and allocation of independent computing branches and related operations requires large labor costs, and it is not always possible to parallelize arbitrary algorithms in general. It should be noted that all existing methods of increasing productivity in PNS have a general disadvantage: the impossibility of parsing the maximum algorithms that are solved at the level of elementary operations.
However, this approach does not always solve the problem of cardinal increase in speed and reliability of performing arithmetic operations in the PNS.
To date, there has been a gap between the increasing requirements for improving the performance of real-time computer systems, on the one hand, and the impossibility of satisfying these requests based on the use of existing PNS, on the other hand.
This fact led to the need to find ways to increase productivity, for example, based on the use of new structural solutions in the creation of CSC.
Scientific researches were conducted in recent years, identify promising ways to improve the performance of computer systems, which are based on the use of the modular number system (MNS) [4][5][6]. However, in existing researches little attention is paid to issues devoted to the implementation of positional operations in the MNS [5][6][7]. This article focuses on solving this problem.

2.
Research methodology and analysis of results

Search of ways of increase in realibility
Currently, intensive searches are underway to improve the efficiency of arithmetic operations through the development and implementation of reliable and fast real-time CSC.
The results of the studies devoted to the improvement of the characteristics of CSC showed that one really practical direction is the approach based on the use of MNS codes [2][3][4]. Ascending from the known Chinese remainder theorem (the task of restoring the original number k A by the aggregating of its remains (deductions) {a } i by dividing it into a series of natural numbers 12 , , ..., n m m m (modules) of MNS) , which was previously interpreted as a structural theorem of abstract algebra, guaranteed the specified parallelism in the calculations over integers, under the conditions that the result of ring operations belongs to the range of integers, defined by modules product of MNS. Having its ideological roots of the classical works of Euler, Gauss and Chebyshev on the theory of comparisons, MNS introduced new ideas in the development of creation methods of highly-productive and ultra reliable CSC [8][9][10][11].
At present again interest in use of MNS as tool for increase in productivity, reliability, survivability and also reliability of calculations of computer systems increases. It is caused primarily by the following circumstances: the emergence of the numerous scientific and theoretical publications devoted to the theory and practice of the computer systems and components creating in MNS [12]; wide distribution of mobile processors that require high speed data processing at low energy consumption; the lack of inter-bits transfers during arithmetic operations of addition and multiplication of numbers in MNS allows to reduce energy consumption; strong interest to MNS is being shown by the banking structures, where it is necessary in real time to handle large amount of data safely and reliably, i.e. they are required highly-productive means for highly reliable computing with errors selfcorrection, that is typical to the MNS codes; -the elements density increasing on a single chip doesn't always allow to perform a complete and qualitative testing; in this case there is an increasing importance of providing failover operation of CSC; the need for the use of the specialized CSC to perform a large number of operations on vectors, which require high-speed performance of integer addition and multiplication operations (matrix multiplication problems, the problems of the scalar product of vectors, Fourier transformation, etc.) [13][14][15][16]; the widespread introduction of microelectronics into all spheres of human activity significantly increased relevance and importance of previously rare, and now so massive scientific and practical problems, as a digital signal and image processing, image recognition, cryptography, multi-bit data processing and storage, etc.; this circumstance requires enormous computing resources being in excess of the existing possibilities;

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the current level of microelectronics development is coming to its limits from the point of view of productive provision and reliability of existing and future computer systems and components of large data sets processing in real time; taking it over nanoelectronics, molecular electronics, micromechanics, bioelectronics, optical, optoelectronic and photonic computers and others are still rather far from the real industrial production and employment.
the modern development of integrated circuit technology allows to have a fresh look at the principles of devices construction with modular arithmetic employment and provides wide opportunities to use new design techniques (such as the methodology of systems design on a chip-SoC) both in the development of individual computing units, and computer systems in general; integral technology enables more flexible design of computer systems and components and allows us to implement MNS -based devices as effectively as on the basis of the binary system; furthermore at present in order to improve the effectiveness of computer devices development, automated design systems (ADS) are widely used; in this respect, the design of computer systems and components based on MNS does not differ from the working with the help of ADS data of binary data-blocks in PNS; unfortunately, Ukraine today in contrast to the theoretical development, technologically is behind the foreign microelectronics of some leading countries; in this case, it is advisable to use the existing theoretical achievements and practical experience in the creation of effective computer systems and components in MNS [17][18][19][20].
One of the disadvantages of MNS is that there are no simple signs of the output of the result of operations outside the operating range [0, М), where: At the heart of the majority of control methods of data are based on the analysis of information, that is on comparison of data. Therefore researches and development of mathematical models, methods and algorithms of comparison of numbers in MNS is an important and relevant task. Now it is possible to allocate three groups of methods of comparison of numbers in MNS [23][24][25].
The first group includes methods of direct comparison, based on the conversion of numbers ) and their further comparison on the basis of use of binary position adders.
The second group of methods includes methods based on the principle of zeroing. The procedure for the zeroing process consists in transition from initial number: presented to MNS to the number of species: The number zeroing is performed in the same way:  [24][25][26][27]. To the third group of methods, we will assign the methods based on the definition (allocation) or the formation of special features, the so-called positional features of the non-positional code.
To detect errors in MCC, the most commonly used procedure is zeroing. The essence of the procedure consists in the successive subtraction from the initial number: The purpose of this article is the development and research of the error control method in MNS based on the application of the zeroing procedure.

Method of errors control
In general, the essence of the procedure of the process of zeroing consists of the sequence of the following operations.
Stage 1. Initial checked number: is successively reduced to the form: Next, the initial checked number:  ) and the operation of selecting the next zeroing constant: According to the values of () , ; , nn a a a a  , etc. After every two subtractions, one additional time step is required to form the next address and access the accumulator of zeroing constants. In this regard, for every two addition clock cycles ( 0 add   ) there is one clock cycle that is free from addition. Let's compare the effectiveness of the method of error detection in the MNS proposed in the article with the existing method based on the procedure of ordinary zeroing.
To quantify the effectiveness of the proposed method, we introduce the notion of an efficiency coefficient: where jnumber of the zeroing method ( 2 j  , for pairwise zeroing; 3 j  , for pairwise zeroing with prefetching of digits; 4 j  , for pairwise number zeroing with prefetching of digits).
Expression (3) can also be represented in the form (4): In accordance with the expression (4), we define the quantitative value () ) of CSC. The resulting calculated data will be placed in Table 1.  Table 1 shows the calculated data  Table 2.  Next, we define the difference ( ( 1) 1,0,0,1, 4 А  has no errors (see Table 2).

Conclusion
The main direction of modern science and technology is the development and use of new advanced information technologies based on the extensive use of computer systems and components (CSC). Information technologies are increasingly invading our lives, penetrating all processes (social, economic, political). Scales and complexity of the tasks solved by modern computer systems impose qualitatively new requirements to their main characteristics: productivity, reliability and efficiency of systems that causes need of improvement existing and creations of new means of information processing. In modern computer systems, an improvement in one group of quality indicators, for example, an increase in productivity, leads to a deterioration in othersa complication of structure, an increase in cost, a decrease in reliability, etc. [1].
In the modern world rapid growth of volumes of information and increase in complexity of the set scientific and technical tasks, connected with achievement of appropriate level of quality and reliability of transmitted data is observed. Therefore, the main objective of scientists in the field is Therefore the main objective of scientists in the field is development of theoretical bases for construction of high-speed and reliable CSC [31][32].
In PNS the problem of increase in reliability and productivity can't be effectively solved without deterioration some key technical and economic indicators of CSC. At the same time, there are positive results of researches which have shown efficiency of application of MNS for increase in speed of realization of integer arithmetic operations, reduction of time of error detection and as a result increase the productivity and reliability of CSC [36-39]. The methodological basis for building a CSC in the MNS involves a comprehensive solution to the problem of increasing the productivity and integrity of the processing of integer data, as well as providing information security, impedance, performance and durability of the functioning of CSC. Existing data comparison method in MNS don't provide the maximum accuracy of comparison of numbers. Thus, there is a problem of improvement of a method of the fast comparison of data based on the application of the zeroing procedure [30][31][32][33].
It is known that considerable time of control of data reduces overall effectiveness of application of CSC in MNS, at realization of integer arithmetic and other modular operations. Results of a research of control methods of the data in MNS which are carried out in article have shown that the existing control methods of data in MNS based on use of application of the zeroing procedure reduce control time [34][35][36][37].
Applications of this method provides obtaining reliable result of control of data in MNS. By the accuracy of the control data in the MNS, we understand the probability of obtaining the true result of the control operation data presented in the MNS [37-41].
The essence of the method of error control is to use the procedure of pair number zeroing with the preliminary selection of digits. This makes it possible to increase the efficiency of the procedure for data zeroing in comparison with other control methods up to 30%. The practical significance of the results obtained is that, in comparison with the existing methods of error control in MNS, the error detection time is more than halved. This circumstance makes it possible to increase the overall efficiency of the use of MNS in the creation of CSC.