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ON-LINE TESTING OF COMPUTING CIRCUITS AT APPROXIMATE DATA PROCESSING
DROZD A.
Department of Computers, Odessa State Polytechnic University, Odessa, Ukraine, e-mail: Drozd@ukr.net
Abstract. Processing of the approximate data has the features essentially changing conditions for the on-line testing ofcomputing circuits. A significant part of the errors produced by faults of the computing circuits does not reduce reliability ofcalculations results. The on-line testing methods show new property that consists in rejection of authentic results. It reduces reliability ofa checking. The on-line testing methods reducing probability of rej ection of authentic results are offered.
1. Introduction
In the on-line testing there were uniform requirements to checking methods of computing and control circuits. These requirements were formulated in the theory of self-checking circuits. The main thesis consists in detection of each fault of the given class of faults at occurrence of the first error [1, 2]. The self-checking circuits of logical and arithmetic devices using parity and residue check techniques were developed [3 -7]. These methods provide diagnosing of a fault “stuck-at 0” or “stuck-at 1” on the first error and, as a whole, have high detection probability of typical faults for computing circuits.
However processing of the approximate data essentially distinguishes computing circuits from control schemes. On the one hand, rounding of the data may lead to calculation of non-authentic results on the faultless circuit. On the other hand, authentic results may be obtained on the faulty circuit because of loss of errors at the rounding of numbers. These conditions reduce efficiency of the on-line testing methods satisfying traditional requirements.
Insection 2 the approximate calculations features that influence the on-line testing methods of computing circuits are examined.
The factors lowering influence of computing circuit faults on reliability of calculations results are defined in section 3.
The appearance probability of an essential error is estimated in section 4.
In section 5 the problem of reliability estimation for an on-line testing method is solved.
In section 6 ways of rise of reliability for an on-line testing method are offered at processing the approximate data. The on-line testing methods with the increased reliability are considered in section 7.
2. Features of approximate calculations
Approximate calculations have the following features that are caused by rounding of the data:
- deleting of rightmosts of the calculated result;
- data processing in the extended formats;
- matching of exponents of numbers.
The first feature of approximate calculations is following from a theory of errors. According to this theory the amount of
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exact bits of result does not exceed amount of exact bits of an operand. Therefore basic formats for floating point arithmetic are formats with single precision.
Processing of approximate numbers necessarily contains the operation of multiplying, as this operation is present at record of number. For example, a floating-point number A = mNE, where m - a mantissa, N - a base number acceptedby default, E - the exponent. The binary operation ofmultiplying doubles a digit capacity of an operand. Rightmosts ofa complete product are inexact and are discarded indataformats withsingle precision.
The second feature of approximate calculations is connected with violation of the associative law of arithmetic operations for the approximate data.
For example, it is required to sum one million with one million units by implementing binary operations and n-bit grid for representation of mantissas, n < 20. Addition of one million with unit will make the result equal to one million, as unit is lost at a matching of exponents. One million such operations also will make the result equal to the first number (one million). For obtaining correct result it is necessary to change the order of execution of operations. First of all the pairs of units are summarized, then results of the previous calculations are summarized. The first is summarized in the last operation of addition.
For restoring the associative law the digit capacity of a mantissa is increased by usage of the extended data formats.
The third feature of approximate calculations is connected to a matching of exponents of numbers.
The matching of exponents is fulfilled in frequently made operations: at addition, subtraction and matching ofnumbers. The mantissa of number with the smaller exponent is shifted to the right with loss of rightmosts. Then rightmosts of results of all prior operations are eliminated from calculations.
Thus, the approximate data are handled with the enlarged digit capacity ofmantissas. Results ofcalculations are rounded off with loss of rightmosts.
3. The factors lowering influence of faults on reliability of result
Definition. The error produced by a fault of the computing circuit, is named essential error if reduces amount of exact bits of result, and it is named unessential otherwise.
The share of essential errors intotal of errors is reduced under operation of the following factors:
- exception of errors at discarding bits of result;
- increase of a share of unessential errors at usage of the extended formats;
- exception of errors as a result of prior operations at a matching of exponents.
4. An estimation of appearance probability of an essential error
The listed factors can be parsed as operating independently. Therefore the appearance probability of an essential error can be estimated a share of essential errors in total of errors by the formula KT = KjK2K3, where Kj is the coefficient which is taking into account lowering of a share of essential errors at the expense of loss of errors in discarded bits; K2 is the coefficient that takes into account lowering a share of essential errors owing to increase of a digit capacity of a mantissa; K 3
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is the coefficient that is taking into account lowering of a share of essential errors at a matching of exponents of numbers.
Coefficient Kj defines a share of errors that remains after exception of errors in discarded bits of result. It is estimated by the following formula:
Kj = (1 -nD/nc) = nL/nc, Kj < 1,
where nD is amount of mantissa bits discarded after rounding; nc is amount of calculated bits of a result; nL = nc - nD is amount of left bits of a mantissa.
At execution of binary operations nD = n, nL = n, nc = 2n, Kc = 0,5 (fig. 1).
1 ... n+1 ...
n 2n
* IlL w . Ki = . ^ 11D ^
^ “C w
Fig. 1. Definition of coefficient Kj for binary operations
coefficient K2 is defined by the following formula (fig. 2): K2 = nT /n, K2 < 1, where nT is amount of exact bits of a mantissa; n is the enlarged digit capacity of a mantissa.
1 ... nT +1 ... n
TTT
K2 = * n
Fig. 2. Definition of coefficient K2
For example, formats for floating point arithmetic ofpersonal computers allow to increase a digit capacity of mantissas 2,7 times from 24 bits in the single format up to 64 bits in the double extended format. It defines coefficient K2 = 1/2,7 = 0,37.
coefficient K3 can be estimated, taking into account, that the matching of exponents is fulfilled by shift of a mantissa ofnumberwiththe smaller exponent. Shift ofa n -bit mantissa on d positions leads to discarding d from 2n bits of operands. Therefore, it is eliminated on the average 0,5d / n bits from all subproducts, prior operations of a matching of exponents, and an error in these bits. Then coefficient K3 is defined by the formula K3 = 1 -0,58Ocd/n, where 8Oc = Oc /O0; Oc is amount of the equipment of the computing circuits, prior shifter ofmantissas; O0 is total ofthe equipment ofthe computing circuits.
If all values of the d are equiprobable, its average value makes half of digit capacity of a mantissa: d = n / 2 , and the coefficient is calculated as: K3 = 1 - 0,258Oc.
For the several ofthe exponents matching operations fulfilled in given computing circuits, coefficient K3 is defined as product of the coefficients retrieved for each of operations. For example, for the adder ofpair products Oc = 3, O0 = 5, K3 = 0,85. For K1 = 0,5 and K2 = 0,35 KT = 0,15 .
5. Reliability estimation of an on-line testing method
The method is not authentic only in case of faulty computing circuits under conditions:
- a skip of an essential error;
- an authentic result rejection executing at detection of a unessential error.
Thenthe metric of non-reliability (complement ofthe reliability to unit) and reliability of a method can be estimated by the following fo rmulas:
dcm = (1 _ Pco )(pskip + preject ); (1)
dm = 1 _ (1 _ PcO )(PSKIP + preject ), (2)
where PSKIP is skip probability ofofan essential error; PREJEcT is rejection probability of authentic result; Pco is correct operation probability ofthe computingcircuit; (1 - Pco) isincorrectoperation probability of the computing circuit.
At an estimation of probability PSKIP it is necessary to take into account such events as generating by a fault of an essential error and non-detection of an error. Generating of an unessential error by a fault and detection of an error are necessary to take into account for probability PREJEcT . Probabilities of these events are accordingly KT , (1 - PD), (1 - Kt) and PD . Thenprobabilities PSKIP and PREJEcT are defined by the following formulas:
Pskip = Kt (1 - Pd); (3)
PREJEcT = (1 _ KT )PD , (4)
At execution of exact calculations KT = 1. In this case the formula (2) withthe registration (3) and (4) will be transformed to a known estimation of reliability [8]:
D M = 1 _ (1 _ PcO )PD .
For Kt < 1 formula (2) shows new property of a method. This property consists in rejection of authentic results of approximate calculations. For exact calculations authentic results may be recognized erratic only by faulty check circuit.
The maximum of reliability DM is achieved at exception of anyone from unauthenticity conditions of a method: skip of an essential error or rejection of authentic result. In practice it is important to limit manifestation of each of these disadvantages. Therefore the formula (2) is supplemented with two conditions:
PSKIP - PSKIP T ; (5)
preject - preject , (6)
where pskip t and PREJEcT T are tolerated values PSKIP and preject , accordingly.
The requirement (5) to tolerated value pskipt has developed in conditions of model of exact calculations for coefficient
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Kt = 1 at which for a selected method with probability PD of error detection it is fulfilled PsKip = 1 - PD .
The requirement(6) to toleratedvalue preject t atallwas absent, as for K T = 1 probability PREJECT = 0, according to (4).
Usage of the same method in processing the approximate data (for Kt < 1) will lead, it agrees (3), to lowering probability PsKip in Kt 1 times. However a new property ofa method to reject authentic results with probability PREJECT ^ 1 will be shown at PD ^ 1 and Kt ^ 0 .
For example, let the residue checking method with probability PD = 0,9 is used for exact calculations (Kt = 1). Then, according to (3), (4) and (1), PsKip = 0,1, PREJECT = 0 and dcm = 0,1(1 _ PCO ) .
For processing the approximate data with probability of appearance of essential error Kt = 1 same residue checking method provides, according to (3), (4) and (1), PsKip = 0,01,
PREJECT = 0,81 and DCM = 0,82(1 - Pco).
Proof. Let PsKip = 1 - PD is value of skip probability of an essential error at exact calculations; Pskip a = Kt (1 - Pd a) is value of skip probability of an essential error at approximate calculations; Pskip a ^ Pskip . Then
Pda > 1 -(1 -Pd)/Kt, (7)
For Pd a > 0 the inequality (7) is fulfilled always if its right part
1 -(1 -Pd)/Kt <0. (8)
The inequality (8) is fulfilled under the following condition: Pd + Kt ^ 1 •
The rejection probability of authentic results
PREJECT = (1 _ K T ) PD A
is reduced at the expense of decrease of detection probability of essential errors before value pda .Q.E.D.
The inequality (7) is a common condition which limits skip probability of an essential error to the value achieved at exact calculations.
The metric of unauthenticity for the method is increased more than 8 times.
6. Ways of reliability rise for the on-line testing methods
From the analysis (3) and (4) possibility of lowering of probability PREJECT follows at the expense of increase PsKip . For this purpose two ways may be used:
- increase of probability Kt ;
The decrease task of error detection probability is solved within the framework of threshold and probability concepts.
The threshold concept consists in errors detection during solution of the computing task, count of amount of errors and matching ofthis amount withthresholdvalue SE . Ifthreshold SE is exceeded, the last error is considered essential. Within the framework of the threshold concept the on-line testing methods are used with probability PD ^ 1.
- decrease of probability PD .
The first way may be carried out at the expense of increase of coefficient K j by decrease of discarded bits amount in result at usage of the truncated arithmetic operations. Techniques of the truncated execution of operations should be considered as a main method of mantissas processing in base formats of floating point arithmetic.
T runcation of calculations for binary operations almost twice reduces amount of the equipment of iterative array computers (multipliers, dividers, shifters) and raises their speed without lowering single accuracy of calculations. The amount of discarded bits is decreased many times from n bits up to nD = log2 n bis. Then nL = n, nC = n + log2 n , Kj = n/(n + log2 n).
The coefficient Kj and probability Kt are increased in 2n/(n + log2 n) « 2 times. Rise Kt from 0,1 up to 0,2 reduces probability PREJECT on 12,5 % from 0,9 PD up to 0,8 PD. Probability PsKip is increased twice, but remains in 5 times less incomparisonwithexact calculations. The metric of unauthenticity is reduced on 10,8 %.
The second way allows to reduce probability PREJECT not increasing probability PsKip above the value achieved at exact calculations.
Theorem. For PD + Kt < 1 probability of rejection of authentic results may be reduced by usage of as much as small probability PDA of error detection without increasing of skip probability of an essential error above the value achieved at exact calculations.
Let Kt is appearance probability of an essential error then 1 - Kt is appearance probability of one unessential error, and the value (1 -Kt)Se is probability ofevent, when all SE ofthe detected errors are unessential (for PD = 1). Then the skip probability of an essential error is defined by the following formula PsKip = 1 -(1 -Kt)Se, and for Kt ^0 formula becomes simpler: pskip = Kt se .
The probability of rejection of authentic results is reduced in SE time and is equal
PREJECT _ (1 KT)PD /SE . (10)
Then requirements (5) and (6) are fulfilled with the registration (9) and (10) under conditions, accordingly
K T SE < PS
(1 -Kt)Pd/Se <Pr
defining choice of threshold value SE on the following inequality (1 _Kt)Pd /prejectt - se - pskip t /Kt .
in the probability concept essential and unessential errors differ at the expense ofvarious probability oftheir appearance: Kt(1 -pco) and (1 -Kt)(1 -Pro).
The theorem is proved, that the errors having various probability of appearance as better differ, as less probability of their detection PD [9].
Therefore within the framework of the probability concept the on-line testing methods with low probability of error detection are used.
Probability PD is limited to time of detection of an essential error that is defined by the formula [9]: T = ln 2 / PD.
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On-line testing methods that detect errors in dependence on their value have additional possibilities for rise of reliability. Let Pd e (We ) and Pd u (Wu ) are probabilities of detection of an essential and unessential arithmetic errors, Pde(We)>Pdu(Wu), We and WU are values of these errors, IWe I > IWu I. Then (3) and (4) will be transformed to formulas
Pskip = KT (1 _ PD E ) ; Preject = (1 _KT ) PD U ,
that concurrently define low probabilities PskiP and Preject at the expense of high probability Pd e (We ) and low probability, accordingly.
7. On-line testing methods for processing the approximate data
The main on-line testing method for computing circuits is the residue checking. it shows high effectiveness at execution of exact calculations and as it was shown on an example, has low reliability at processing of the approximate data. The residue checkingmethod that is adapted forprocessingofthe approximate data by generalization on a case of the truncated arithmetic operations is offered. Onthe one hand, truncation of calculations raises reliability of a method by increase of coefficient K T . On the other hand the method promotes wide usage of the truncated operations, that for iterative array circuits essentially reduces amount of the equipment and raises speed. The further rise of reliability of a method canbe fulfilled inview of highprobability Pd withinthe framework ofthe threshold concept. The method has received practical confirmation. The devices developed on the generalized residue checking method, are recognized as inventions and implemented in a batch production [10, 11]. The method is applied forthe truncated operations of multiplying [12, 13], addition[14, 15] and division[16]. Withinthe framework of the probability concept some methods of the functional diagnosing are developed. There are methods on the false-code of arithmetic operations results and a method on the limited set of the entry words. They provide high reliability at the expense of lowprobability of detectionofunessential errors [17]. Methods ofa logarithmic check [18], a check oninequalities [19] and serial check of items [20] additionally raise reliability, fulfilling error detection in dependence on their value.
Conclusions
The fulfilled researches have shown essential influence of the approximate data processing on methods of the on-line testing. Methods show new property what consists in rejection of authentic results. The obtained estimations of method reliability have shown low efficiency of used on-line testing methods. Ways of reliability rise for on-line testing methods are defined within the framework of threshold and probability concepts. The generalized method of residue checking adapted to requirements of the approximate data processing is offered. The on-line testing methods raising reliability within the framework of the probability concept are developed.
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