BSTJ 40: 2. March 1961: On the Construction of Minimally Redundant Reliable System Designs. (Ray-Chaudhuri, D.K.)

6.7 MB·English

On the Construction of Minimally
Redundant Reliable System Designs

By BD. K, RAY-CHATIDHTRT
(tanunsin sine Seeman 7,180)

Siveral authors wr conideral the pueibiity of Inewasing the ve
lty of ange and comply bonny Agia eyteme by antracing come
esata in te atv. Ta 9 rompenin pope, Armatoug! prspoate e
Ieeme for opplaing error cuttin tn apnehatn digital system. fo
this paper tee evclop a general matbematial teary for generating
sity secant mecractning oe for the atk ie gation Th
reels tv aha! are ead “wnsnimaliy wean rls eptems.” The
room of constructing mama redundent reine apne wl ote
Dus dan of vce wo Gere fst Sn ma ene vk ef the stom
Se compelly med An example is conatderet vm detest showing how the
theater oa te 2

In eunples binary gia stom eroplang large suber of blake
ui ghetsiea eqapme often ix dill a enrune safe eel
‘elility of eae single Block equipment. Aw alec tn allan the
ese gues of revsility by senprosing the sei af each Boe
hag: prev to no vel etter handy itm sore

feelundaney ia the syser, i is wvsble to eonteuct highly eeliable
Capleton, eve though ote single lore nok aa high rahe
Moorea Shvimone Tigwn’ Won Newnan! Coggeen cl Anson

have considenal the pwiblen of souetructing sebable syrem designe
Fri paper geal anatheratral teary ts bre evel fa he
ronisttsion af saintly redunciot ellesweter deigar bee ea
Uvewene tied by Armstrong Ths theory lool vlan to the
vary of ervor-corsvel ng lee. The rubies if consti miimelly
edna sytem esi seer jae wl le foo ance whee
nei ant ft rst one blk of the ysis complete ened ie
‘hi paper
596° nu wz, NAA HreUNTEAT RUSK, WARE 1

Spee thee ae. inary ie vavnbles X), Nay ==25Xy Tet By
denole the set nf 2" mepace binary sequen, Beery ae of wales oF
Whew Lig input vial il be egurded aes wane of By Ane
Trapping of to wil beled a Bowens fein uf the m inp

ashlee Xy, Xe, sym. Hoe the sake of brevity, the cllertion of
put watabies willbe denotes ay. Lot
Sn Pinyin finns © Sany — Fe Pies See

se pt Hoole fantom of the m Kinsey wacables Xi Ney o> hm
(Gur relies in canst systems whieh wil iene he pe
Boole fanetins wi highilgrewut rll. Thesgtem weeebloks
felted equipments each af which ean agzitesan » Boolean fare
tions. For the sake of hesity. a elletion a p Donk functions, wll
Ta called Loolan p-ftetion. Thus J. = (Ja yfuy "fig is Booenn
prltmetion. Any Rasleun p-fonoion iw mapping of By inco By. ac
‘ork of our srwem eyathesisce « Boon ptusttion, Fig. i «oche=
nati igrom for the original nonredundant system,

"The books wet of unis in thw eyotem. If there i «Seu ix bleak,
‘thea some oral she p eusputs of thr block are erraeous. Ln other words,
inthe ease ofa ual « Dock wil syuthesae the corresponding Boolean
pplunction song's. Het Pe! vote Ue oot 2 ey pple, Chon
tty Bovlear petro ‘aes vals on Vs Ua Var devote Ue rt

‘ach cy 8 an clement of Ty
‘Lat f f.). Thon f ean be regarded oe a
hpi i no, We dts an of pcp se
‘ual mod 2adution. Furesanpie, ify ~ 3, ey ~ (001) snd as ~ (UCL),

I c+ ae = (100). Lar ado! be oa serene of Py! given by
f= nya eh and a= aad oon ah Whe ae Fa
fs define to the clement fay + ay a6 fh The p-tuple
Tie) wil be calle the wid clemone of V2, The might aa of
the Keverion a is defined :o Ie the muster won leazeats among
uve, aes For any partie value 2" ef the input veeabiee
Ha?) lea veer iV, Suppose thee ate faules in t(¢< #) blogs
‘Thon ¢of ta fncions fff wll seutheizedsronely. Lece
Ahn pat wil be the weston f°) fg where e = Cente sh
vector in Ville weight 1. While designing « systert to syuthesine the
rola fanotion (ome mighe eequce Ghat whenever the mumber 0
fanlty blocks is or Kass theourpt ie orortr, ue en adie Thi
introucng moe nssdaney in the syste, hy myescing (e+ 2}
{Boolean p-anctius and ailing 2 gia corrector nit vo the system.

Suppese es ress. 6n6 1 ole panels wal i mapping
ot Verona Ve West onaider @ gy gay “oor am function
from Bato V2. Fie every value X? of Xy (in an elena of Py
Suppose te Funeione and (pone the property P stated below

Ror viny seesor ¢ Selpnging to," wit aCe ot exceeding ¢ sd
‘every wale A” of the ina variable

Cup6XO + = 8 o

‘he fwselions ¢ and © enable

vs fo sonteuct @ sytem which wil

yn the Boolean palit tiai Jos ee of eur whenever
tie muster af tly Blacks in the we #2 lee. ‘The n Boolean
Tuesioutvs seat he emedesed ata colletion ap Bunter,

functions of mnt vaviaes, ‘Therefore we ean evlly obiain the
Tegal desi of wnystan which wil sethesse there np Boolean Fun
tions This sgatem wll be callin Ube evilor swbayaton, Silay, che
function € can be considered sn elect ofp Boal fuxetions of
fp binary it varus ad therfore we ean obtain w system Which
Sei acheize howe pt furl. This sprtera wil be eae the oor
ect vsligatem The wp cuits of the end subsyetem wil be tar

“sof tho eanretor subsystem. Now it easly een that, because
fhe progeriy of the fant g etd Mbensver the mime of
fant bloc the enor ule i 8 and che erator it
ittiee nf erer he a tate af the sormtr sgt wel br

FIN = EIR FAN
XI SAN eat Naat Ne ier
Fol Sl We =a)

A schematic dngran Sar sho whole ester geen i Pig. 2. view of
508 ume ne, svar rmeteAT parnVKn, Na meHE “et

meu py,
i |
L am 9
==
tho abore dseuson. the ling wwe fins given Tel are

weigh

Dentin The tonetiont eye, 64 and C possemsing the
nopestyP stated fa 1) ills riled eels stm dengn of order
[ea essary e = — doe Pho Boeloan tances fey fi

Definiin 22 A reliable system design a! ardor ¢ and sedimeaney 7
fr the # Boles pfunesions fy, fo, ~~ fe wil clin animadty ve
tute it tho vedundarey #2 eee veliabe system design of order
"iv the scane Suction is ot lise thar

Inthe pret per ne lve mien el of ining ily

redtaat syst ceo ee ar any a of Baer
for sebitary # od p-Hyetens desigue of ligher weer wl Le given ia

sales pe
‘We fuave tnd fe redundancy ras amas af the extra atnount of
ounipstent shih has sv be toe Got taking te agrlees celle td
Tron ne sek che set whivh has mint posible vale of the ge
Aivtaney ns epi ot that ne acamaa thae the rorrestae
Pulasatet dewet mae ate ern at all "Therefor, tonne be whole
“leveopmient pracy Houle, Hs impel al ite he ae
ff equipment nevswary forthe rant ulnystew fe anal in ome
prison tothe arnt equipment ees fe the whale ap¥cem, oF
That other segs I taken asa suggested in Ref hy to ens
stilt of te worn ete. We have nol une sy a inmalial
aeporne his roquiement. i the develapment of the

thors

Consider two vectors wen a Ieloaging 0 Vy" The dteare dfoya"s
botieen the tivo elms of Fy ied € lela +o}. Tht
Daespssua enrspasey mekzasta waa 30

= Logynon eos em ot 20
She msn of steers for which on 2 a4 = 1p p Forexanspe
Ip n= shana = (01,10) id 9? — (01,1610), then ao =
{Wio1.W) wl ane?) ~ 21 cam be ewily eheokes chet the ditenee
Aisin) bow -atcties the tones emeitinns ol aidan fanetion, We
hoe men Ms Belean pation etd i Sertion IL ea be oe
ihre ae moppinge of fy ante Vy A Boolean pfunetion fy wil be
led a dagenanat Buulor Saveton for any elezaeat a of Vy
Tne iva eahue of the inst yards X tor wih 20) — a. We
‘hall testi that all Boolean pfusetions sppeasing i uur discussion
te nondegenerate, Inthe hilowing we bavee = 2 and nF

"Pheo f) A neeeesty wi sufr'srt coaitinn that there ets
seliahie syste design of omer # and reducdaney » for thn & Booka
Dnctions fu Jey Joi ths these exists a east A uf V2 wont
tng element sich thal a) 2 2C-F eed © As a

‘Poof Neca, Suspose wow ev ai 9 welll svat Ar 0
font ther Tanetions be — tye) a the coer
Fhoeson fe. Faw every walt X? of che input variable X, gf") 3 8
retro! Vy", Cowes the st

Aa tei) XG Me

Using the fact thot the Raakan pfuneions fe, «25 are nomen
igeorate Fantions, it follows rasly that the set 8 corn nT
Srecnra af yh Conder tw diinet wekom ral a of he eet TE
posible euppess lla’) 2% Sineo ni?) 2) wean find vector
Cot Vpteuth tinh ate = a? teddies) ££ Sine alel Sly we
have

bun = Cal be ane a)
[Bquation (8) vovlendiess the assumption thal » nd a! are dstinet
Yeetone of "Poi spect the root ef acne

Siow, Buoynws le eubect of V7 comtaing # element aaa
Dein the property that ne?) & BE Is aya! yw 2 as Wo ast
tp tu-towne eorrespunnone Iewern the # weetore of Py and the
oF sroters fl, or evere value ot the pau variates WON) =
Toe gaN hts NOD fea seta of Vad shore es umreoling
‘rotara af Fy Delongingio."Thnenehefnetiong = (ecvses os.)
ieee by

ER = BET 3 ee AI ee HN

besa ta wo
WoO ee nina svstune necestoen aoesn, AREA at

shore cis the wert oH elgg A and worrespcng 19 Te
Sector (LX°) of Ve ectoreo function © ie deine in eh felling
rmaniee. Tet y ='G1,, 74) boat asbitrary veetor of Vy" Fist
We ehoeen a vector w belonging to a auch that dima) lr,
toe! (lc Lat = (8,9, yo Oe che woetr of Vy" whi eorseeponde
fhe. Thea we deine

co) =a oy

Thuy Cs manping of Fe onto Py ist tr sherk that the en

oder function g ad the canrsererfunetion C defined chore ports the

tomporry stated in Setion LL. ‘This eamplstes the proof of sue-aney.

‘Tieaam 2 Li shore ests a robb spam design of order fund re-
tundancy » far F Bones peirtiony ee

vert (Je-ve()e-vt+Q)e-n, w

where a = ef rand
Profs Feo Thoorea I it is ueewsney thal there existe awubeet A oh
with the propery that
Mog?) BBA
lat Sy done the set of vesters > of Vy" with the property that
Aya # tI follows cay from (7) that, For any tn dstinet vectors
fend of of A, the sete S, end $," do nol nave any zomsnan eement
Tot 8. denote the sot of elemnts of Vj? which huve distance from
y= O12. Obviously Sais tho union ofthe (1 + 1) ate Se;
Bayo 8. canenine

od A oat fi

+ ()u=0

sori, Thre ate sh nmovesapping sts and the tote nombor of

Skomigar sie ome eto ‘Oo}.

eee[io(u-n+ (Qua

‘Theorem 2 follows fom (83
yet design of one ter Baolesn punetions, Theorem 2 ie wetually
"ecnceabzation of felt uf Hamming

Tiel nar) dene dhe axis integer » for whieh dee existe a
relalenyster dep ade andl sodudaney for Fn =e Bohan
spfuncions. Port ~ 1, the inoquality (6) hovers

aeons

Hen we ave

1 ection V we sal show tha

1 ere exits role eyo ssa a onder¢ andl redundancy for &
Panes pofunetions, then mite) =

Tem

wet me

of: Suppose me) = 2 Thon here ests teliableayste design
ofr Cand redandaney rfor& = w+ Boolean punetions. Hence
bay ‘Tlwonean Lthsce exit a suet fof Vy" contaiing st eloments
wwith the property that aa’) 2 24 Lj mae yw ew Torney
Proteus ooo ol We ase the weetor

= fanart

of Po Tine we have a subset dof Ve
lls poses te property that el na
Hea, hy Theorem 1 ve ca asin a oiable apstem design of order ¢
nun! seundaey nor P= 69-11 follows tba

ni Shed Lent dame s

heorem d> I fr celiahleavetea design of onder (and redundaney
for Tnlean ptanetinns we bse

dr =e eR Smt

ry

Hh te design e mivilly ronda,
‘Peni Ii poedle, sippie the ayetem isnot mixizally redundant
‘Tim then existe a ohuble sytem design of onder ¢ and muna”
UZ em nena SwRI rreeNteAt SOUHNAE, MARCHE 11

Tor Bune pedis ben ig sna pain iter
emi? 2) 2b} es, BY Lemme |

nr Bee be

EEL @-0, 9)

The Foquality (10) mesos Tae igual (Hs heme the theonem
“all

Th thi seeton we sll wonder pseu subelas of spat di

sigue hefner og deine, To define che ina ester =
Sips we ne tase te tery fn el a A be sh ne Hele
Ml edoaerinty 2 roving = 2° elemwnts aed ¢ denote a pinion

cement af Ks binary ple (
resp tae elem

Neyo dps) wll be made to
PP of K and

Vie vent. Aw ment a = ayy ay oy
Tider am an aevector with olexieuty i Kia wight we) of a io
Sal to che nmber of nos elements onion oy: Me
cum of fo vectars ae = fe ug, +a) eda = fe

defined toh

Fal Co bit my osm
Obinusly V," is 8 eter spe aver K. Conider w ayitom design fr 8
rules pfutetions Spo te ener i ti i

‘For every vove 3° of the input variable N,
68) = WM WT] or (8
i veetor belonging to "Let
Am Bia iX eB. ay
Depwitiow 3: A eystem disige for Bowl
loa system xin ifthe subset of Vs died by (11) fo wertor
Terwen 82 8 snenaary aad alfvent condition that teliale ion
system sign fie © Boal fone ions eof onder #38 that hee
‘Meng soma. eeetor of the svt fined I HT ul law
Geb.
‘Profs Heese of Uhre, i woul beeen to how tha

Maal) 221 wal eA, aaa ay

e2 p-imitions is si ob
listeria BROMNDANT MULAAMLE shspi8a8 603

By detaiton afc) = ae — a) Sine A en woetor spice, bo!
‘eee element off nd shy ®— ao” ha tonal element of 1, Henee
follows that (12) wl hold aad only ole) 2 2-F T foe every nan
a leant

"Dajte fo A nals A with lets wl be id to Fave Pe
tPorpeoperty fo msl the assy ae ines depend

Theor $7 A nesoaty and saivient condition for te existe of
telihle linea system desig of ver Fand rekduney efor & Boolean
Pimesione is ual Hiercanile x aig Mish a= (i br roe andr
felts ith ermeata in F wich posses (Por

Profs Slfeeney. pose the tne Wise iy

pe | Mame tn

Tet eno the vector spur orthogonal to the vector sper ens
by the rotary verlors af Ms roan at Inst elementos Te ould
Se ation to sbow thn Ue weight of ay toma ses A ea
fact Ge -F Li IL posible, suppore 1 cantare s soneul erotor ith
Sweight Inc haw (20-4 1. For szpliity of wring sane that the
Neon = dims aes ane O.=-.D)denystn wera ya ys
Ere tone ests of Kher en

a, fgets Hoes Howey OF Moy)

Bquation 14) imple thot the fist 2¢weetans of the wsileie I ane
Tivonlyckpendest chi Te contnaditier Tie wompleti the pn

slices, Novowity ean he prowed by exsely sila arguments
"he wader aequsinted ih the hleatune ox eror-oorreting fnene
revi wal teengaze from Tense ttt sealer ete
Sigu af der for Borde efonetions exists al only if Crm
egrets with mph ae afr

‘omoetng ina eat

fio pees existe, Leman and Thongem 4 een above ate not new
sean Hey were pee Bose” asa Zine in w eierent “orm. Wo
Enve inl sr! pis fr these reat for te ke wpe

In 4 sceton yo sha hoe Is fe sstreting zininelly oe
dundantfiwar syetem desis fone Afar oon ptt fr
say sabienry yale Cael p
Tern 8

nde) =
cass

Poses Ln Setion IL we proved ta

mo a2

ene it wil he stent show te
Hl

nian tt

«ast

Tr prowe (12) sve shall construct a mnsris M ithe eoumns and x =
oF = 1)jta D sows hich has (P2)-poperty. We shall denote the elr=
entaof by Oty, +n, where Othe pul elements i the
rukiplseive entity. Conviler she mie AF ven be

["]
a= cas)
L

shone isthe identity nitric with rows wl eulane ad Bf 8 8
snstrix with rensms wad # (m7) roe given below:

Doo 1

an

oo 0

bake 2

1 can be easily cela thot he matriy A hus (P':)-propert ie m0
hyo rows of APs Koratly dysnent, “Tis wont Une pron! of
Th ebwull be observed hal Tlnorein 6 enables wb csteus
wally rsndontsystom designs of onder I for any arbitrary vals of ke
nde Far gv fe dwt ta integer + for whi
air <e
mG) ~~ bw somseruc the amas AY with eas and (7)
tune dete in (16) ad he an the atom desig a trae