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MICROCOPY RESOLUTION TEST CHARTNATIONAL BUREAU Of STANDARDS- 1963-A

ONE-TEP DIAGNOSIS ALGORITHMS FO)RTHE BQM SYSTEM LEVEL FAULT MODEL

G.G.L. Meyer

Report JHU/EECS -62/14

mcctrical E umnern and Cocnptuer Scoe DeparunmcThe Johns Hopkins University

Batimore, Marylandi 21218

Deeme 15, 1962 < .C D

*'JUN 16 1983

11l wrwk *rn supre byteOfxo aa Reearch under CtRurSm

N=44O00

-2-

V ABSrRAC'r

A mu-step r-fault dkagnosable system is a system in which all faults may

be kiemified from the test results, provided that the number of faults does not

exceed r. In this paper we present two algorithms that may be used for the

one-step diagnosability of the system level fault model proposed by\liars

Grandoni and Maestrin ) The first algorithm may be used when the system is

one-step 7-fault diagnosable and no two units test each other, and the second

alorithm t be used whenever the system is on-step 7-fault diagnosable.

/ W

SAccesssion For

Avet&.i-

t o r1t

1. INTRODUCTON

In the pat few years a grea deal of effort has been devoted to the presen-

tation and analysis of system fault model& Theae models ame characterized by

the facn that they do not empitasn test generation, but rather emphasize the

use of test results for the purpose of fault loottion and Vault detection. The

problems associated wit system level fault models are (i) presntuation and

desciption of the fault models themse1ves 00i amaysis of the dkpgosability

propwats of the fault model under various assumptions concerninig the faidt-

test relationships and the system testing Interconnection network and (iii) sy,-

thesis of diagnosis algorithms for fault detection, orn-step diansis and

sequenial diagnosis.

Many fakult. models presently exist [2], [71, [8], (91, (14], andi their diagn-

ability Properties aft essentially well undemsood (41, [51,1[15-17]. The state of

fth art, as far as one-step diagnosis algorithms are concerned, is less satisfac-

tory. The two most widely analyzed system level fault models are the PMC

model, prqxosd by Prepazata, Metze and Chien [ 141, and the BGM model pro-.

posed by lBsrsl, Gramloni and Macatrini (21. There exist algorithms for the

one-step diagnosis of the PMC model; unfortunately, they suffer from some

serious dmwbackLc The algorithms proposed by the author and Masson, [101,

(121 ame only applimble to regular testing imteroonniction networks and the

Mins proposed by the author (111, (131 have been shown to work only when

the first Hakimi-Amint hypothesis [5, Theorem 11 is stisfied and the number of

faults is lkmhed. The Algorithm proosed by Corluhan and Hakeml (31 depends

ont an unproved cornlctume The algorithm proposed, by Allan, Kameda and

Talda [11, (61 involve aee marching a&Waccacig

-4,

The situation for the one-step diagnosis of the BGM model is ev= less

mtisfateory. no algorithm has been proposed to date. To remedy this sination,

this paper presents two algorithms for the one-step diagnosis of the BGM

model. The first identifies the set of faulty devices, provided that the first

lIkkkni-Amin condition is satisfied andi the set of faulty modules is no greater

than i, wher is the minimum number of system units that test each system

unit. The second identifies the set of faulty modules whenever the system is

one-step fault diagnosable. These two algorithms are easy to implement and

their very existeno is an a pseaori indication of the strength of the asump-

tions of the BGM model

In the first part of the paper, the BGM model is described. Then the basic

tenet of our approach, i.e., the partition of the system units into four subsets is

presented, together with the properties of that partition. The first algorithm

malms direct use of the basic partition properties, and its deoding properties

are exhibited. Finally we perform an analysis of the failure mode of the first

algorithm and present a remedy - the second algorithm. We then show that

this algorithm may be used to decade syndromes for the BGM model, provided

that the model is otr-step r-fault diagnosable, and that the number of faulty

modules is no greater than r.

2. THE BGM SYSTEM LEVEL FAULT MODEL

The BGM model proposed by Blrsi, GranConi and Maestrini [21 is a sys-

tem Ievl fault model closely related to the PMC model proposed by Prmparata,

Metze and Chien [14. In the DGM model, a system S consists of n modules

U0, U 1, .. , U,,- and a testing interconection design TD, where

TLD-((iJ) U, tests

It is asumed that no module tests Itelf, Le., the diagoml does not belong to

TIDL A module I assmd to be either faulty or nonfaulty, and the state of

each module in S is assumed to be constant during the appLimtlon of the testins

prowdums. If (iJ) is in TID, hen U, tests Uj, and the test outcmne alj Is

assu ed to be either V1' (Uj passes the test) or " (Uj fil the test). The set

of test outcomes I a,., I (1,J) E TID ) is the syndrome of the systean. In the

BOM model, the following relationships between fault and test outcomes are

asume&

(i) if (IJ) is in TIDand fU, and Uj are nonfaulty, then aij - 0;

(ii) if (,j) is in TID, U, is nonfaulty ald Uj is faulty, then aij - 1;

(iii) ir (J) is in TID and both UO and j are faulty, then ai - 1;

S(iv) if O) is in TID, U, is faulty and Oj is nonfaulty, then aj may tak either

the value 0 or 1.

Thus, the main dilfrem between the PMC model and the BGM model is that

if a module U, is tested by a module Uj, and if aj., -0 , then the module U is

nonfaulty.

A fault s ftttlt of tim stem S i dombed by the set Fs of the faulty

modules in & A set of possible sy dromes coresponds to each fault situation.

Given a fault sitation, the mmputation of the cowr srm s is not

difot, but to cwnpte th fault situations that am consistent with a given

smdnmum is not as eas. In this paper, we address the latter probm -

zmeiy, decoding - and %% restrict ourselves to a m-ste i taut

diwsgmublty in the .me of Prepaa, Metine and Chien [141.

IM I-1: A sstem S is onm-step -fault diaosmmble if all faulty modules

-6-'

within the se am be klenifled withot e provided that the

mznber of fhulty modules does not exceed r.

In the remainder of this work, s will be the index set that contaim the

ilc of all the module in S, i.e.,

s - 0,1,2, n-};

fs will be the index set that contains the indices of all the faulty modulo in S,Le.,

fIs- I e s I U, e Fs ;

and given an tndW set a, la I wil be used to denote the number of elements

in a.

3. IMPERFECr ONSTEP DIAGNOSIS

Our appmach to system dkgnsis consists in partitioning the set s Imo four

subets (v, h, / 2 And h3) tht are M y to COMpME and then to rlate thas

four subsets to the set fs which contains the indices of all the faulty modules in

The hxlex set v otaims the indices of all the modules in S that are tested

by at least oln other module In S anti found to be nonfaulty by that module,

I Ea jins exits sothat (J) ETDaIand -Oa . (1)

Thus, I S is a BOM model, i.e., a fault model that satisfies the asmmption of

section 2, the module U, is mnfkulty whenever the WaxlI is in v.

The index seth, contains the indilces of all the modules in S that are

emed by at lu one module Uj,J in v and found faulty, and the indices of all

-7-

the modules in S that test at lut one module U, J in v, and find it faulty, Ue,

hi si E J in exists so that (J) E TLD and aj. - 1U (IE s I J in exit so that (J) E TID and aj - . (2)

Ome should note that ifS is a BGM model, then the index sets v and h,

ar disjibh, and U, is faulty whmer the index I is in hl.

The index set h2 depends on the crdiniity of the sets L(I), where, for

evy ind ins- (v U h1), the sets L( ) are defined by

LO)--JEs- (v Uh) (I j)ETIDa daaj-U uj CJ s -(v Uhj) I J) ETID and ap -}

Gien an index i, it is pomible that an index J exists so that the pairs (ij) and

(,1) ate both in TID and %4 - a - 1. Obviously, in such a cam the index J

appears in LO) only onaL The set L(I) contains the indice of all the modules

adm to the module U, that must be faulty if the module U, is actully non-

faulty. Given a scelar r, the set h2 consists of all the indis in s, but not in v

U It, such that the uerdnalty of LOi) is strictly neate than r, Lae.,•~h ! E, S -€ (V U hd) I ILW ( ;or> + 1 }(3)

It is clear that if S is a DGM model and ifat most r modules in S ae faulty,

then U, is faulty whete I is in h .

The Ibdex set h3 contaim the indices of the runall modules in A L.,

h3 -S -- (v UhlU h).1

In this section w will sine the properties of the sets hl, h2 and h3

when ew.y module is teted by at Iwsr other moduls. We will then ue

those prqpmiles to symheuse a decoding algorithm that produce. an Inex set

fA co'ntn the Indice of ad the faulty modules and some nonfauty oes,

1rovided that the number of faulty modules If , is at mostt. We shall

assume that the following assumption holds.

JHymrea 1: Every module in S is tested by at least r other modules in S.

The next lemma is a direct -onsequere of the properties of the BGM

model and Is given without proof.

Lnmm 1: If S is a BGM model, if Hypothesis I is satisfied and if

Ifs|', then the hdex sets hi, 2, h3 andfs satisfy

hl Uh 2 Qfs Ch l U h2 U h3.

Laemm 2: If S is a BGM model, If Hypothesis I is satisfied and if

Ifsl 1,-1, then IhI .4,- I, - ,h3 - # and

s -hi.

Proof- Let U, be a nonfaulty module. By assumption, every module is tested

by at least . other modules and the fact that Ifs I <r - implies that U, is

tested by at last one nonfaulty module, say Ub. Thus, if U is nonfaulty, an

bxkx j exists so that QI) is in TID and aj., - O, and it follows that v contains

the indices of all the nonfaulty modules in S.

Now let Uj be afaulty module. Hypothesis I and the fact that ms I -1

imply that U is tested by at least one nonfaulty module, my U. Thus, if U, Is

fautty, anlft J in v exists so that (Jl) is in TID and am - 1, and it follows

that h Iconuis the indlices of all the faulty modules in I Ckly, v and

form a partition for , and we may conlude that / 2 aW h3 are empty. o

Lamm 3: If S is a BOM model, if Hypothesis 1 is satisfied and if

Ik U h2 U 3l r , then

fS -h, U h2 U h3.

-9-

Proof. if li U U 0 r, then either 'sI O 1ASI I f

Ifs I <r-- 1, then Lzla 2 implis that fS -hI U h2 U h.

ifIfs I - v', the fat that fs U h2 U h3 implies imed elythatfs

mus beequal to h, U 2 U h3.D

ILZAm 4: If S is a BGM model, if Hypothesis 1 is satisfied, if Ifs I r, and If

Ih1 U h21 - r, then

As - hl U h2 .

Pof If US I1 < r and if IhI U )21 -T, then Lemma impe s imedite

that fSmUt be eqal to hI U h2. 13

We will now show that if IfsI - and lhi Uh 1 I ( 1, th nuMbrof nonfaulty modules i/ 3 is at most 1.

Lamm 5: If S is a BGM model, if Hypothesis I is satisfied, if Ifs I - T', and if

III U h 2 1 -- 1, thenh 3contaitheindicesofatmost r nonfauty

modules and

IhI U 112 U 131 (2.

Proof. If an bdex I in 113 coresponls to a nonfaulty module U1, then U, is

tested only by fauly module.. By assumption, Ifs I - v and

Ih, U h2 iCr -1 an therefore the index J of at least one faulty modult is

in h3. The modul Uj cnnot test mom than r other modules in h3 (otherwe

J wouid be in h I U h2), and it follows that the number of nonfaulty modules in

h3 isumt t'. The indexset U h2 U h3 contaim the ndlem oal the

fault modules and SinM h3 oains at most .nonfaulty modulm,

Ih U 1 2 U h3 4C 2-. 0

- 10-

Lemmas 3 and 4 sumest the followmg decoding algorithm.

Step 0: Compute the at v as in Equation (1).

Step 1: If Is - v I , let fA - s - v and stop; otherwisego to Step 2.

Step 2: Compute the sets h, and h 2 as in Equations (2) and (3).

Step 3: If Ih I U h 2 1 - 1., letf 4 - hz I h 2 and stop; otherwise, go to Stop 46

Step 4 Let fA - s - v and stop.

The properties of Algorithm 1 are easily obtained from Lemmas 3, 4 and

5, and are summarized below.

T/mom . Let S be a BGM model, let Hypothesis I be satisfied, let Ifs I < ,

aid et A b the set genrte g by Agorithm 1. If fA 4 r, then f, - fA,

an if IfA I ;o v+ 1, than If, I - r, fS C fA, ad h f5 <.

In 1974 Hakimi and Amin [5] proposed two character rations of the test-

ing intaeconnection design that insure one-step v-fault diagnombility for the

PMC model. We will now examine the implications of the first of these

assumptions on the sets hl, h2, h3 and Algorithm 1. We begin our investip-

tion by repeating the first Hakimi-Amin assumption.

HypowaIs 2: No two modules in S test each other.

LAna 6: If S is a BGM model, if Hypotheses I and 2 are satisfied, and if US I

< ,, then ether Is -v Iv k - 1 or Ihl U hal - T.

Proof" If Ifs I < - 1, then Lemma 2 implies that fs - s - v, and therefore,

Is - v I v- I.

If IfI I - , then h3 is ether empty or no. If h3 is empty, then

S - II U h 2 and Ihi Uh 2 - . If h3 is nOMpty, then let I be in h 3. The

fact that no two modules test each other Implies that no module in h3 may test

- 11l-

any other module in h3. It follows that U, is tested by exactly r other modules

that must be in k, U h2, and we may condude that |h, U h21 must be equal

to.. v0

The result of Lemma 6 shows that Algorithm I generates the set of faulty

modules in S whenever Hypotheses I and 2 are satisfied and the number of

faulty modules is no greater than v.

Theorem 2. Let S be a BiM model, and let Hypotheses 1 and 2 be satisfied. If

Ifs 1 . v, then Algorithm I stops either in Step I or Step 3 and the set fA that

it generates satisfies

fs -fA "

It is clear from Theorem 2 that if S is a BGM model and if Hypotheses 1 and 2

are satisfied, then S is one-stp v-fault diagnosable.

4L OWE-STE DIAGNOSIS

At this stajM we have a dewding algorithm, Algorithm 1, that produces an

iXex set fA that is eqsl to fs whenver YZA 1 fI ", and that contains the

index of some nonfaulty modules whenevr IA I ;P 'r + I. If, in addition to

Hypothesis 1, we assume that either Ifs I 4Cv -1 or that the first Hakkni-

Amin hypothesis Is satisfied and Ifs I < v. thenf - fs. In this sectio ., we

will modify Algorithm I so that it produces the set of indm of all the faulty

modules when Ifs I v " and the appropriate amumptions on TID are verified.

When fypothesis 1 is satisfied and Ifs I < r, Algorithm I fails to pro-

duceasefA that is equal tof$ when IfsI - r.ald 1h, Uh 2 I <.1. The

only index set that presents a problem In that ame is the set h3. It is Clear that

if the index I is in h3, the module U, is tested by exactly -v other modules

-12-

(otherwewoul beinv U h Uh2). Thus' since Ih U h2 I 4 P- 1, at

least one index J(I) in h3 exsts so that the module Uj(,) tests the module U,.

The fact that the index J(i) is also in h 3 implies immediRty that the modules

Uj() and U, test each other. We are led w define the folowing index set w

that depends only on the int'-connecion design.

DLfkiion 2: Let w be the subset of s that contains all the indices I in s such

that:

(i) the module Uj is tested by exactly other modules, and

(ii) an index J(i ) in s exists such that Uj(,) is tested by exactly r other modules

in S, and U, and Uj(i) test each other.

In view of this discussion, we arrive at the result below.

Laimm 7: IfS is a BGM model, if Hypothesis I is smisfied, if fs I a m eand if

1h1 U h21 <4 , then

h3 Cw.

The set w giew in Definition 2 may contain indices that carrespond to

nonfaulty modules. We must now define a subset x of w that possesses the

desired property: a module U, is faulty whenever I is in h3 r

Doom 3: Let x be the set of all indices I ins such that

(i) U, is tested by exctly r other modules;

(ii) an index J) exists such that Uj(j) is tested by excl r modules in s and

U, and Uj(,) test each othe, and

(ii) an index k() In exs such that the module Uk(,) tam Uj() but nt U1,

and U(j) is tested by at least one module that does not test U1.

The set x given in Deftnition 3 depends only on he testing inmetrcxAaton

- 13-

design; it may be computed once and stored. Its imporance lies in that it may

be ued directly to find faulty module in h3.

Lmwma 8: Let S be a BGM model, let Hypothesis 1 be satisfied, let

Ifs - and let Ih, U h1 -1. If the index I is inh 3 n x, then I is inIs.

ProofP Let i be in h3 n xand let J(/) and k) be indices that satisfy the

assumptions of Deflnition 3. Suppose that U, and Uko) are both nonfaulty.

All the modules that test Uj and those that test Ukci) ur then faulty. The

module U1(k) is tested by at least one module that does not test U aid there-

fore, the assumption that both U, and Uk(.) are nonfkulty implies that at least

r +1 modules are faulty. This is impossible, and thus, if Uj is nonfaulty, Uk(,)

must be faulty. The module Uk(,) does not test the module U1, and the fact

that is in h3 imples that U does not test Uk(i). Thus, ifU, is assumedto be

nonfaulty, we must conclude that at least i' + I modules are faulty. Om

apin, this is impossible Thmfore, the module U, must be faulty - i.e., i is in

fs.0

Using the result of Lemma 8, we may modify Algorithm I to inxporate

the fact that under the approprite colitiom, the set h3 n x is a subset of f8 .

A/pitim 2:"

Step . Compute the set v as in Equation ().

Step 1: If Is -v 1 I -r, letif, - s -vand s top; otherwise, so to Sto 2.

Step 2: Compute the sets h1 and h2 as in Equations (2) and (3).

Sop3: If Ih U h2l - r, t -h U h and stop otherwie, o to Stp 4

Stop 4 Let fB - hl U h2 U (h3 x) and stop.

At this point, it is dar that when the approp to a ptions, am

-14-

atisfied, Algoithm 2 geeates a set.f that is a ko boind for !s.

Lamm 9: If S is a BGM model, if Hypothesis I Is mtied and if

Ifs I < r, then the set f, 1piurted by Algorithm 2 satisfies

In their 1976 pape, Basi, Guandoni and Minstrini [21 proposed a ani-

ton on TID that insures one-step i-fault diagnombility. Using our notation, we

will now repeat that assumption and show that it may be used to insure that f$

is found.

JHypxw8s 3: If the indicas IandJare in wand if U, and U, test each othe,

then IorJ o both are inx.

,mmn. 10. IffS is a BGM model, if Hbpotheses I and 3 are atisfd, if

IUs I - v andif I 1 U h2I - v - 1, then

lh3 n x! -,- Ih u h2I.

Proof: The st h3 Contains at least - IAh U h2I indices and therefore, if allthe itm in A3 M also in A then

Ih3 n xl -Ih1 U h21.

If at lm oer xn , my i, is1nh 3 b nottinx, thenU, is tested by at lmst

v - Ih 2 hya modules wth Ikes hi h3. All the moduies with indik

in h3 that test U1 ar also tested by U,. Hypothesis 3 implies that they are all in

xand therefoe,

Ih3 n xl r- Ih U h21.

LmM 9 knpUiM s that all KM inhl U h2 u (A3 l x)are infs, andwft

the aumpdxon that Ofs I r, we obtain

- 15-

1h3 I x1 -,r- h1 U h21.

The results of Lemmas 3, 4 and 10 show that, when the approimte conwil

dons are satisfied, Alsorithm 2 generates the set of indices of all the faulty

modules in S.

Lewn 1.1: IfS is a BGM model, if Hypotheses I and 3 am satisfied, and if fsI

< r, then the index set fy roduced by Algorithm 2 safis

s -*fa.

Note that Lemma 11 imples that if S is a HGM model that satisfies

Hypotheses I and 3, then S is one-step 7-fault dignble ie., the the result

given in [2, Theomm 2, p. 5951 is retrieved.

It is known that if a BGM model is one-step "-fault diagnable, then

Hyotheses I and 3 are satsfd (2]. -nce, we obtain the main result of the

paper.

Thw em 3: Let S be a one-step T-fault diagnosble BGM fault model and let Fs

•be the set of faulty modules in S. If IFs I v, then the index set fu

crated by Algorithm 2 satisfies

Fs -{ U, j I E fA.

__ ____________________

1.1

-16-

R ERENCES

[1] Allan, F.J., Kameda, T., and Toida, S., An Approach to the Danonsability

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[21 Brsi, F., Graindoni, F., and Maestrini, P., A Theory of Diagnoability of

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(31 Corluhan, A.M., and Hakimi, S.L, On an Algorithm for Identifying Faults

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Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of Ithe abastra entered In Block ". II different be Reps )

1. SUPPLEMENTARY NOTES

IS. KEY WORDS (Coinusme rweroo side I IMeso. amd IdeunIF S IMeek Mmibe)

fault model; diagnosability; fault analysis; system levelfault analysis; one-step diagnosis; syndrome

25. ABSTRACT (Candine m revel e od It nee***" And Idenify Sir bak numi.)

The paper presents an analysis of the B( system level faultmodel of Barsi, Grandoni and Meestrini based on system partition-ing. The properties of the partitioning are then used to synthe-size tvo algorithms for the decoding of the syndromes produced bythe BG model.

DO 14n s17 noN OF I No O is oSmsLETS/N 010- .6 01d 6601 SECURITY CLASSFICATION OF tels PAGE (hm basee,

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