BSTJ 60: 9. November 1981: A Fault-Collapsing Analysis in Sequential Logic Networks. (Chang, S.J.; Breuer, M.A.)

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A Fault-Collapsing Analysis in Sequential Logic
Networks

fy S.J, CHANG and M. A. BREUER™
(Manas cated Seotombor 11, 1680),

Although a sequential circuit Mf reduces to a combinational ne
work Cu after all fesdback paths have been cut, an application of
‘Bovsen and Hong's checkooint inbelimg procedure to Cus does not
necessarily yield @ minimal solution. The set of checkpoints a0 ob
tained will ince ail feedback fine, In this paper, tia shown that
theve feedback fines are nat necessary checkpoints under @ “delay
equivalence” relation tn addition to this, we alzo shove that nat every
Janout branch is a. necessary checkpoint. Any “inpular fanout
branches” can be removed from consideration. The results of unr
analyse lead to a minimal checkpoint labeling protaber for sequen:
Tal lagie networks

1. INTRODUCTION

Becaute there are 3 = 1 possible maitiple stuck faults in a logic
ncework containing w distinc Iocan where cignals may fil (each
Tecation may aemume one of the duee possible etatee normal, stuck:
fn-b, or stuckeat-l), test generation and simlalion prooedures often
reste to fanz collapsing techniques reduce che number of faults
tthich need to be considered, To date reported results in che iterate
Meat only with combinational loge, This prompted nur interests inthe
research to be dizcusted inthis paper.

"Bossen and Hong introduced fvul cllapsing technique for com
binetional logic networks called checkpoint. labeling. procedure.’
‘Checkpoints as defined hy them, are." niece of specified points in
the network such that any mleple Flin the meter is equivalent
to some mulkpl Fault among thee specified checkpoints! The cheek
pointe defined in Ref. area minimal tof poines having the property

sone, tinny 9 Southern Caldera, Le Azgalor
Jost stated. A navoral question iy What ara the checkpoints ina
‘Sequential circuit? Te ix weil known chat by cutting all the feedback
Hines, a sequential network can be mapped into « combinational
network. A reasonable approach wld beth apply Bassen and Hong’
labeling procedure to the zonltng cominsrional metsrork. ‘The check
inc abained would then elie the eedhack lines In tis pape,
tte alow that in goneral all thew fenlhack tinea need at be check
points under a elation enlled “Aelay equivalence” to be defined later.
‘We mill oe that the nober of checkpeins can bo greatly reduce for
a highly sequentil logic network if our results aze utilized. This, in
tum, greatly reduces computational complexities for multiple fault
tnalyat in sequential neewerks. A possible application of our resus
tan be found na paper by Chang, Si, and Breuer:

1, FAULT COLLAPSING IN SEQUENTIAL NETWORKS

‘The checkpoints in a etouit axe specially designated signal lines,
(Checkpoints sre defined wo Uni fur arbitrary stuck-at-faul «there
feria at least one equivalent fll defined on tha checkpoints. Hence,
itehere are w 1 checkpotnts, we need only conser 3 ~ T multiple
faults, Boseen and Hong have developed 2 labeling procedure for
secifying the minimal vt of rheckpoints ia a combinational circuit
"Their resulta seem to be an extension af che work of Shertx and
Poage2™

‘Por convenience, Bosse and Hong’ procedure is as follows

(i) All he primary inputs chat do noe fon out azo eheckpuints
i) Allehefanoul branches ae checkpoins.
Gi) NOT fate are considered as lines.

“Although a sequential cies ¥ rcws ton combinational necwork
Crafter all feedback paths have been cu, Boasen and Hong's labeling
procedure does not necessary yield % minimum number of check

point for such ieuit

‘Consider a synchronous sequential recut M represented by Huff
tmantmodel ax shown on Fig, where Cy ithe combinational portion
Of My and Dis the eet of delay ehanents. We assume that faults in Cyr
tte reetricad to euckeal yp aa Gute in delay’ elements (D f/f)
recut in atuck outputs, Gules and ip flope of M are eonnectod by
ledges, An ee of Mi either line ora branch. Namely, iis either
Primary inp, » primary output, efanout stem, or a fanout branch,
‘leo we shall axe the following:

‘Ascupion [The outpat value of w gabe «function of each of ex
inputs

“Assumption 2: The number of edges in fis finite,

‘Aavumplin 3: There are no "inaccessible" edges in M. An edge of

2280 THE BELL SYSTEM TECIINICAL JOUNINA:, NOVEMBER 1021
Fig. Halon! of sential

_M sail to bo inaewessibe if cis not a primary mpc and, moreover,
Fes not driven by any gabe or memory elements of

‘Almost all practical sequential circuits satisfy che ebove astump:
tions. Figure 2 shows « Hlesidous integrated creat chip. Kage 8 is
nacrenible

‘Restricting our actenrion to snciontfnilts, we dofine a ingle fal
fs exactly one edge stuck-at] (68-1), oF stuck (wend), and a
‘multiple fault an a collection fone er more) of single fully each
fwociated with a diferent eege. Ala, we shall not consider inform
tent fate

‘Be wpplying Boaaen and Hong's procedure oo the combinational
euwutk jn the ant af checkpintao nbeained wl include all Teedbac«
Tines denoted hy the vector ¥. However, we shall show tho Usa ibeet
of checkpoints is not neeostary under a delay-equivalence relation.
“Later we aball consider synchronous sequential vires

1 SEQUENTIAL NETWORK WITH D FLIPFLOPS.
“Two faults wand in M are sid tse dela:equibadene(d-equiva
ent) of order &, if M with u is equivalent to MM wich v efter an

‘application ofan inpur sequence of lenge a least &
Tete specific value of Fis not of intrest cys we shall wey say

Fig 2A tvs mig cea i,
‘that wand wate c-equivalent. To demonstrate this concept, let us refer
to the sequential ereuit shown in Fig. 8. The fault a ad is d-
cnuivatentof order | to tho multiple fault 9 a and e s-4.
emma 1: Any mauttiple fault ime delay flip-flop is dequivalent of
trder | to a stuck input.
Proof, Lat y and Y denote, respectively, the output and input of a
delay f/f. Then, xiq + 1) = Yq) for all g. Thus, a stuck output is de
equivalent of onder 1 to 2 stuck input. Also, a stuck input and output
in deequivalent of order {to 9 stuck input only, and is hasjivatent of
onder 0 to «stuck output ani.

Because of this lima, we shall consider a delay f/f as w 1-input

sate under the doquivalence relation.
etton 2
i) A sequence of edges of M, denote by [sy Si +0584 22+ Bb

where 5 forall t= j,i anid to be w forums path, If for each
Ee neither fs) «ie a fanowe stam and 51 isa fanout branch of 8, oF
{t) there este apace of AF sayg, auch thut sand se: are, respectively,
ti input and the aucpu of 2

Tray = sand a= fi tho path [sy smo ald bo be a forward
path from a taf ffs 8 primary outpat, the path i sad to be a
terminal forward path of

ii) A sequence of edges of M, denoted by [oi fu oo. === tds
‘where s,s fr ll be said co be a backward path, ior wach
Pe either (a) sis. fanone branch and its am is, or (0) there
fists gnte of A, suy Bauch that and oy, are, respectively, the
fusput and an inp of

= cand s. = 8 the path is aid to be @ backward path fram «
tof IE Bina primacy input, then the sequence [8 ==> 184 == 5 So
isan tobe w terminal backesard path of

I |

4g Aopen det

2262 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981
“A backward path from lB donated by i said to hea backward
oth of wi tere exists cge Tin Af such tha a backward pach
roma oT enneine every element ofp be path ail o terminate
a8

Glealy 8a primary input uf M, then buclowaed pals Fon or
lo A isalto a terminal buckovard put of

‘Th path (8+ fy) Read to paws throug wach 5. Conversely
‘each 5 id lo he contained in the path. Branch fn alka said to be
an element of the path [31 s+ Sa:

‘Asan exsmple, consider BF on Pig. 4. The sequences [12,15 6.88
411, 20] and (12, 17,10, 5,8, 9.11, 30] are forward pacha of edge 12.
lowover, the sequence [12.18 6,8 20, 12 16,1, 18,3. 9, 11,20]
‘nea forward path of 15 becmse wie 13 ae A appr lsc nthe
requence. All backward paths of edge 29 are shown in the grap of Fig
5. Note th this yrupl is tn with edge 20 as the eho, ad che
Teaves are echer primary ils or Fat Uncles

Detinton 3
“every age of M pocessaesateat on terminal frst, he
(Mica fo he a regular sequential exci, Otherwise BY si bbe
regatar.
‘We wil for our avin om repularsequest ial crnit

A sensitized path from an edge = x to un ou:put fin Fis 2
terminal orwaed pach from oo f slong mith constant signal walacs
trsgned co some ofthe other edges such that charging Uhe hie vale
tall change the lagi value of 2
kraut fee 2 Line naan no inary

In the itcuit chown in Fig. 4, we can sensitize the odge 1 wo the
onipul 21 hy setting edge 9 to sat and edge 13 to -t.

Teonma 2 Let g be agate in Ma regular sequential creuit, and x1,
sy cov od be the input tog Forel t,thone bau eaten mali
fault in AE aghich wil semstise x loc nd

Proof: Sineo M is ropalar there existe ut leust one vermin forward
pac foe. an usp, Using the conve uf Beutler,
‘the condition for sonst the output of gL che ing: bs given by
tho equation

Me a0 ge = gabe bh

Be moa mb+ Be
where

HOE He Be Dt eet
and

Bem lay oe te Botan tah

Now select any minterm of oor fg tis mintorm specifies,
snewhere mie ithar O or 1, thon leche x input cog be suck ala, Nowe
‘Soume thav for Un: ternal forward path from ilo B, he tout of
{gi an inpul iu gate A. Repeating the prove just devctbed, can be
Senstizod tog. This chaining pros bs entinved uncle reached.

etrton 5

Tet wand ff be bwo eyes af M, Bdge a ie sud lo dominate edge 2,
ithodh » rac and wa! will cause Af to become independent of the
Sgnal on vegerdlos of whether or aot other alges are normal

‘2204 TIE ACLL SYSTEM TECHNIGAL JOURNAL, NOVEMBER 1581
‘To dlustrte this, let ws consider Fi. 2, Buje B duminabes edges
ind 8. Faye 7 donninates all other edgea. However, edge 2 does noe
Mlmninate eget, Recsune sehen edge 48-1 and edge dal, edge Tis
ts fuoetion of edge | independent of che state of edge 2
"Thenrem Rage a dominates edge ff and onty if all terminat
faruard paths of Bcontuin w
Proof: Suppose all forward puths uf 8 pass Unrough a. 8 stuck « will
bloc any sign on "Thus, Af hecrmes independent off us sel.
‘Therefore, lerrinates f. Suppo there exist one terminal forward
pra of f thal does not cone ‘hen. by Lemma 2 one can alerape
find a suukiple fault on 4, whieh wil enstio tne signal on # oat
leaac one ofthe primary oucpats of M. Thus, M's dependent on and
se daes not deminate Tie proves thr only if par
"Theorem 2 Dominance relation inclaces«parlin ordering.
Proof: Bvery ford path of a passes through a. Thus, « duminales
fe Tel ay deainate a ad og deiinese iy. BY Theorem 1, all forward
pacha of as and a: Fa55 chroush as and a, respective
forward path ofc also pes throngh mT pen Liat: a
flominntes ay Suppose a dominates # and f dominates «Hy Theorem
{yall forward paths of pass chrough «and al forvard paths of «puss
‘ecg Thin ean that fr fone of « and foe » forward
path Thevefore, « dominates {and ff dominates a implite a = 8.
‘Nan; dominance relation is veflexve, ceansitive, and antiserametzc
We conclude dha it induces a partial ordering

Tia combinational network, « fanout brunch never deri other
fanovt branches ofits stem. However. in sequential cieuit favo
tranch may dominate other fonout branches of ie stem. Consider, for
‘nag, the seuentalcneuit Mf, which inatiown on Fig. 4, Branches
‘ant ave fanou branches of stem & Since every forward pach of 10
pases chrough 9, branch 9 dominates branch 10,

Detintion &

‘A fanout brane Uhl dovsinsother fanout branches af te stem
nid toe stuewar, otherwise, i sa vo he mesg

orem 4: li) A sfem possesses of mest one singular branch
i) Beery ningudar Branch dominates a atherfanout branches 2f
Proof: Lar 8 sem which ponnaos singular roars le dna by
tnd ie fanout branches be Gerioued iy er, yoy tm, were a ik
singular. By definition, cy must dominate we Teast one ar (42). I ay
seer Lor i as singel Hen oy wl eaten ne Tora
rath chat does not pass Unrash es hecaes every forwenel path af
fontaina the fenout stom 2. Thuy, wll ew longer be dorainated
tin Thiseonieuliedin praves (- Suponse senue as oe aoninated by

FAULT-COLLAPSING ANALYSIS, 2966
so, Thea, ag poseosos al Tuas one forward path That dows nov pass
Uhrowgh ay. IC follows that efor ull # 1 also possesses wt louse one
farsa pi Chat dos st ps Uhrough oy. Par a) of the Uheorem
fellows asa eault of Theorea 1

etiiton 7

"The ferme sow edge a of M, denoved by Kia), ia at of edger of
‘Mauch thar

1 every backward path of » contains exacely one member of Kd,

(G) members of Ala) are either primary inputs that do not fanout
or nonsingular fanout branches, and

liiyevery foreard path from each member of K(a) co a does not
onlin ny oer nonwingula fant ranches

‘Let ud consider again (eee Hig} All backward paths of ely
are shown on Fig. 8 Each underined edge is either a primary input or
forwingulee fanout brasch Every starved edge ia an element of
the kernel sec of edge 20 [Le K(20)}. Prom Fig. 5 we have £20) =
(14,13, 14,36 17,18). "he following kornel acts an be easly verified:
UU) = UJ, £2) = ©, 10); and KGS) — [4 16,18, 17, 18
“Lemma's: Let a and bbe eny poirof edges of M.

THA stuck ai ceayuivaden! to sume madple ful emmy ite
eral set Kis)

(G8 1f Kio) ane KE) are disjoint, chen any muttipte fault among a
‘and fe sheouivalent f9 some multiple fault among Kia) UK.
‘Proof Pax: (i fellows jointly from the fact that amy mtipe feuls of
{a loge gate ee eel multe ules searing che nat nes
tly he deiiton of Ka) whieh requires thar every backware
path of « contains exactly one member of Kia). Part (fellows part
1a) of the lems.
emma If a does not dominate then either (there exists no
Forward path from fi to a, or (2) every forward path from 8 lo @
‘contains at leart ane nonainguar Branch,
roof acaternant (2 iat, thou a doos aoe dominate.

Suppose there exist a forward path from f 0 a, denoted by» =
Irs, +++ au: that does not contain ny nonsingular branch Them ss
cher afanout fico stem ora singular branch, Lt srbe singular. Then,
12-4 fino! slo. By Theorem 8, a other fanout branches af 8,
fe dominate Dy se Therefore, ll forward pats of passthrough
+ Thus, all forward paths off aleo pace through a, This implies that
tt domtinsiesf. Suppse every sy ika stem without fanout, chen pis
fanont foe pa, Bul by Theoret 1, Unie lo ineplies that er doin
{Tho contradiction proves the ler.

“Theorems 4 Tf asues nol dninute ond 8 dos not dominate o, then
(hei herned wet tre inn

2286 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1881
Proofs Tat 4 ¢ Kip). Then, 0 falls into exactly ane of the folowing
valgus
i) He in a backward pall uff which does nox contin any ee

in common with any backward pach of

{i i concn in» fran path fom 2

(Gib o @ enntained tn a bacicwand path nf 9 7, that does not
contain «hu paid a backward ph of, say 2, eins common
Iaranches

Clearly, ite belongs la caegony (wn Ku) fo elon eategore
(Gd, hy Lemma and category (lof the definition of Kel, 6 Kio
So, comtiler only ategore i. Lat p= [2h a, 000 Suh and g =
[ects v-+ tok Catagory (id) implies that here exists w anv such
at, =f and af forall <a, 2. Namely, (ort) iw nowt
stem Thyra! &-, are nonsingular, by definition o Lies im forward
path from hy c0 that by ne? Rta). If «Se sguler, by Theorem &
Feo snot. Ph, brah fia dominated By yp, and so 0 Is 932
flnninsted by f then 20 #o. This eonteaits te assmption. There-
fore, ao tt slontinaced by fy Loria 4, there exits =~ 2
uch that oy iy 2 nonsingular branch, This implies thet « i
Therefore, Kf) at Kiet are dso.
“Theorem 6; In « regular sequential cirexit, every singutar branch is
nota necessary thecigwint, but al nonsingular Fano branches cre
rneceseary checkpoints
‘Proof lit fine fanout sem wth funoul ranches ay ++ dy whore
‘Brig aingulan By TTheorens 3 avery tc wy i dominated by oy Thus,
lohuck eedgs is cLequivalent toa etc ft ege Aa toa stack a eee,
for any ## Tri i not chequivalent to either a stuck 8 edgo, or stuck
fu edge, or suck oy edge fory 1, Therefore, all nonsingulur faut
branche uf fare necescary checkyuins
Theorem 8:Ror any iregular sequentied circuit Mf there exists a
regula sequential eieuit 0° such that Mand Mf" are equivalent,
‘ead furthermore, M preserves janl! hebaviar of M under shuct9>e
Fath axsuanptin.
‘Proof Being iveguley, Mf poasesees to wets of tyes, donated by
{and 1", wher each member 9 dots nol hive any terminal forward
paths, and enc snember of &” posceses at leas: one termina] forwuel
path, Tfolloe thar no primary orl of Wis fanetion of ce signals
pr any member of under faul-free or any auck-rype fault eandilions.
"Therefore. omuoving all members of & will not alter the normal or
normal fneionel behavior of Af Tha rising ceca consists of
only edges in S* and ia ropa

‘Forlluteate chs theorem, consider sequencal circuit Man Fig 6a
Cone can wnslly verify chat @ ~ (4.5.8 11, 18,14, 16) and 8 —
{L288 7, 8,8, M12, 16, 17). Afr removing i fom Af, and

AULT-COLLAPSING ANALYSIS 2967
Hg bn a Tern Ss a) Che
‘labeling, ane obeaina che aoquental circuit Ma, shor on Fig Uy
whieh ie setnllyw eumbinntional ee
Theorem 7: In sequential circuit M, either (i) the hernel sek for any
edge in M exists or (i) M is erequtar.
Proof: Suppo benn edgeot BEL a ackwnrd path of be denated
boy p= [ey se oes sto here 9) =, Then, 2 belongs la one of the
following eatorories,

Tip iva ternal Talwar pth; sony, 8 primary inp of

M.
iA There enita +< w auch that 6 and ¢, axe fanout branches of
+ (See Fig. 7a)

‘ih 'Theve ern o gave g uch thac ais an input of g and sy is che
conta of g. (See Fig. 76

Ie) i nnocessible,

Category it) never occurs because of Assumption 3.

‘Suppoce p fala into either entegory (or Ui. Then cvery backward
peth ofa contains nt lest one nonsigula Cau raneh oF a primary
Impat. This implics that Xia} docs eis if ease (ii) dues nut wpply

INow consider ons (0). Soppan there exits no a 1 © that a,
fant ranch, This reson il very elemento does 01 hav
‘ermal forward path. Thay Mi ieee

‘Suppnse there exec some I= £2, thot ina fanout branch. I,

2260 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981