Logic Redundancy Identication. September 14, target specic faults and are sometimes referred to as. \fault-independent methods.

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1 Fix-Vlu n Stm Unosrvility Thorms or Loi Runny Intition Sptmr 14, 2003 Astrt { Thr is lss o implition-s mthos tht intiy loi runny rom iruit topoloy n without ny primry input ssinmnt. Ths mthos r lss omplx thn utomti tst pttrn nrtion (ATPG) ut intiy only sust o ll runnis. This ppr provis nw rsults to nlr this sust. Contriutions r x-vlu thorm n two thorms on nout stm unosrvility. W rprsnt sinl ontrollilitis n osrvilitis usin n implition rph n its trnsitiv losur (TC). Both omplt n prtil implitions r inlu. Wknsss o this prour r in lin with th ts o xvlu vrils on TC n th lk o osrvility rltions ross nouts. Th x-vlu thorm s unonitionl s rom ll vrils to th x vril n thn romputs TC rursivly until no nw x nos r oun. Th stm unosrvility thorms trmin th osrvility sttus o nout stm rom its omintor st, whih ithr hs x vlus, or is unosrvl. Rsults r onsirly improv rom th prviously rport implition-s intirs. In th 5315 iruit w intiy 58 out o 59 runnt ults. All 34 runnt ults o 6288 r inti. 1. Introution Runny rmovl nsurs tht iruit os not hv unnssry loi ts, whih inrs th hip r, powr onsumption n proption ly [2, 18]. Anothr rson why runnis shoul inti is th rliility o th iruit. In th prsn o runnt ults som tstl ults my msk. Thr r thr irnt pprohs to runny intition: utomti tst pttrn nrtion (ATPG) [4, 5, 8, 12, 15, 16, 23, 25, 27, 28], tstility nlysis whr pnin on th typ o mtho w us, i.., xt or pproximt, w n intiy ll or sust o \proly" runnt ults [9, 20, 22, 26], n implition-s mthos [1, 7, 11, 13, 18, 19]. ATPG mthos n n ll runnt ults ut thy hv vry hih omplxity. Tstility nlysis, otn rstrit to linr omplxity, n intiy \hr to tst" ults som o whih my runnt. Implitions mthos o not urnt to n ll runnt ults n thir omplxity is polynomil. Both tstility nlysis n implition-s lorithms o not trt spi ults n r somtims rrr to s \ult-inpnnt mthos." Chkrhr t l. [3] n Booln ls untion to onstrut n implition rph rom ivn t lvl ntlist. Both tru n ls sttus o th sinl lin is rprsnt y nos in th implition rph. Pir-wis rltionships o sinl lins provi implition s in th rph. Sinl pnnis riv rom th trnsitiv losur r us to ru trnry rltions to inry rltions tht in turn ynmilly upt th trnsitiv losur. Whn th ATPG prolm is pos s Booln stisility, th implition rph provis usul hlp in solvin it. Rnt work hs prou svrl ATPG prorms tht us Booln stisility [4, 12, 16, 27, 28]. Th implition rph n trnsitiv losur hv lso n us in ult-inpnnt pprohs to runny intition [1]. Hr osrvilitis o sinls r rprsnt s itionl Booln vrils in th implition rph. Gur t l. [6, 7] improv suh runny intition thniqu, usin sust o ll prtil implitions riv rom th hihr orr trms rprsnt usin nin nos. Mht t l. [18] implmnt ll th unimplmnt irt n prtil implitions n this improv th rsults onsirly. Th nos in th rph rprsnt th tru n ls sttus o th sinl lins n tru n ls sttus o th osrvility o thos sinl lins. Dirt s rprsnt th Booln implition rltionships twn nos. Th motivtion or th prsnt work oms rom th wknsss o th xistin thniqu. An nlysis o runnt ults oun y natpg prorm tht oul not inti y th implition mtho shows: Th iruit topoloy ors som sinls to x vlus. Th implition rph prours oul not nlyz th t whn st o x sinls inun th stt o nothr sinl. Thr is no nrlly known prour to trmin th osrvility sttus o nout stm vn whn th nout rnhs wr unosrvl or h x vlus. Prpr or DATE'04 P1o6

2 Amon th ontriutors to th non-rph s implition pprohs r Iyr n Armovii [13], Pn t l. [21], n Zho t l. [29]. In Stion 2, thorm or x-vlu nos is ivn. Stm unosrvility thorms r isuss in Stion 3. Stion 4 proposs improvmnts in th rph thniqu s on th Booln ontrpositivlw. Limittions o our thniqu r ivn in Stion 5. Rsults r isuss in Stion 6. Stion 7 prsnts th onlusion o this ppr. 2 Fix-Vlu Nos A Booln vril x is rprsnt ytwo nos x n x in th implition rph. Eh no n in tru o ls stt. Whn x =1,nox is tru n x is ls, n vi-vrs. Th vril x ssums x vlu 0 i thr xists n x! x in th implition rph. Similrly, x ssums x vlu 1 whn th x! x xists. Whn svrl nos hv x-vlus, w hv oun som limittions o th prviously sri trnsitiv losur thniqu [6] in intiyin rtin runnis. W onsir: A primry output AND or NOR t with n unxitl s--1 runnt ult on its output lin (s s--1 in Fiur 1). A primry output OR or NAND t with n unxitl s--0 runnt ult on its output lin. Th ollowin thorm intis th ov mntion runnis. Thorm 2.1 (Fix-vlu) I Booln vril in th implition rph is x to tru (ls) vlu thn thr xist unonitionl s rom ll othr nos in th rph to th no rprsntin th tru (ls) stt o th x vril. Proo: Consir x sinl, = 0. Thn, thr xists n rom no to in th rph (whr n rprsnt tru n omplmnt vlus o sinl lin in th iruit), thn irrsptiv o th sttus o ll othr nos in th rph, will unonitionlly tru. This \unonitionl truth" must impli y ll vrils. In othr wors, thr must s rom ll th nos in th rph to th no. Th xmpl iruit shown in Fiur 1 hs 7 runnt ults. Prvious work [6] oul intiy s--1 ults on lins n. Thy oul not intiy th othr runnt ults: s--1 on lin us = = 1 i not imply = 1 in th rph, n s--0 n s--1 on lins n wr not inti u to lk o stm unosrvility rltionships. Without Thorm 2.1, w oul not intiy s--1 on lin s runnt. Fiur 2 shows prt o th implition rph. Th s othr thn th thr outoin Fiur 1: Exmpl iruit with x-vlu nos. Fiur 2: A prt o implition rph tht intis = 1 usin Thorm 2.1. s rom xist in th trnsitiv losur. Th ults s--1 n s--1 r inti to runnt u to s rom nos n in th trnsitiv losur rph to nos n, rsptivly. Ths ults r lssi- s unxitl runnt ults. As ths nos r x to loi vlu 1, orin to Thorm 2.1, w unonitionl s rom ll othr nos in th rph to nos n. This inlus two s rom to n, rsptivly. Th trnsitiv losur is otin y rph trvrsl rom h no to trmin its rhility. Trvrsl pros throuh n nin no (ll s ^ in Fiur 2) only whn pths rriv throuh ll inomin s. In this s, thror, whn w strt t, is rh n th! is. This sts to x vlu 1 n intis th s--1 ult on lin s runnt. Th othr our ults, n s--0 n s--1, r not yt inti u to th sn o ny ls stm osrvility s. Ths ults r inti- s runnt with th thorms introu in th nxt stion. 3 Stm Unosrvility Thorms W n th omintor st o nout stm in iruit s miniml st o sinls throuh whih ll orwr pths oriintin t th stm must pss. Trivilly, omintor st or ny sinl n oun s th st o rhl primry outputs (POs). A omintor st onsistin o sinl sinl is ll uniqu omintor. Our nition is onsistnt with th rph thory [10] n hs lso n us in n ATPG lorithm [14]. Thorm 3.1 (Stm unosrvility) A nout stm is unosrvl, i h sinl in its omintor st ssums onstnt vlu n: th nout stm os not hol onstnt vlu, or th nout stm hols onstnt vlu n, in spit Prpr or DATE'04 P2o6

3 o ny lol hn in th stm sinl, th omintor st vlus o not hn. A lol hn o sinl only ts th portion o th iruit twn tht sinl n POs. Proo: From th nition o th omintor st, i th stm unr onsirtion is unosrvl t th omintor st thn it will unosrvl t primry outputs s wll. So: i (X; x p )= (1) whr X is th st o primry inputs, x p is th nout stm unr onsirtion, i is th omintor st untion, tht is untion o X n x p n i 2 [1;k], whr k is th rinlity o th omintor st n is onstnt, whih n ssum loi 0 or 1 vlu. Expnin Eqution 1 usin Shnnon's xpnsion thorm [2], w t: x p i (X; 1) + x p i (X; 0) = (2) W onsir two ss: x p 6= onstnt, thn Eqution 2 implis: i (X; 1) = i (X; 0) = (3) Tkin th Booln irn o omintor untions with rspt to x p,w t: From Eqution i (X; x p p = i (X; 1) i (X; 0) (4) Thus, x p is i (X; x p p = =0 (5) x p = onstnt. Sustitutin x p = 1 in Eqution 2, w t: i (X; 1) = n i (X; 0) = unknown (6) n sustitutin x p = 0 in Eqution 2, w t: i (X; 1) = unknown n i (X; 0) = (7) Th Booln irn o o omintor untions with rspt to x p i (X; x p p = i (X; 1) i (X; 0) 6= 0 (8) In th spil s, whn i (X; 1) = i (X; 0) =, i.., i (X; x p ) rmins unhn whn x p ssums two irnt vlus, Eqution 8 vluts to 0 n th stm x p oms unosrvl. unos. stm 2 1 unos. omintor unos. omintor (ix) os. stm (ix) 0(ix) Fiur 3: An illustrtiv xmpl o Thorm 3.2. unos. stm 1 2 1(ix) omintor Fiur 4: An illustrtiv xmpl o Thorm 3.1. Thorm 3.2 (Stm unosrvility) A nout stm is unosrvl, i h sinl in its omintor st is unosrvl n: th stm os not hol onstnt vlu, or th stm hols onstnt vlu n, in spit o ny lol hn in th stm sinl, th unosrvl sttus o th omintor st rmins unhn. Proo: To provi in th nl sumission o th ppr. Th nout stm, shown in Fiur 3, is not x n th omintor hs onstnt vlu o 0. Thus, Thorm 3.1 intis nout stm s unosrvl. Th nout stm, shown in Fiur 3, is x to 0 n th omintor st ; hols onstnt vlu o 0,0. Thus, Thorm 3.1, lolly hns th vlu o n hks th t on th omintor st, whih is oun to hn. Thus, Thorm 3.1 os not intiy s n unosrvl stm. Th nout stm, shown in th sm ur, is not x n th omintor st sinls ; r x to 0. Thus, Thorm 3.2 intis s n unosrvl nout stm. Th nout stm, shown in Fiur 4, is not x n th omintor is x to 1. Thus, Thorm 3.1 intis s n unosrvl nout stm. Th nout stm, shown in Fiur 5, is x to 0 n th omintor st sinls 1;2 r unosrvl. A hn in th vlu o hns th osrvility sttus o th omintor st. Thus, Thorm 3.2 os not intiy s n unosrvl stm. Thorm 3.2 is onsistnt with lmm ivn y Iyr n Armovii [13]. Thir lmm is spil s o th Thorm 3.2 us th thorm os not rquir th unontrollility initor onition rquir y th lmm. Atr th osrvility sttus o nout stm is i, n unosrvility is implmnt in th implition rph. Prpr or DATE'04 P3o6

4 Fiur 5: A iruit with osrvl n unosrvl stms (ll ults shown r runnt) Fiur 6: Exmpl iruit showin th intition o runnt ult s--0 u to ontrpositiv s. 4 Usin th Contrpositiv Lw Aorin to th ontrpositiv lw, i thr xists n rom no p to no q thn thr must lso xist n rom no q to p [17, 24, 29]. Th us o this onpt urthr inrss th numr o runnt ults inti in omintionl loi iruits, whih is vint rom th xmpl iruit o Fiur 6. Fiur 7 shows th trnsitiv losur rph or th iruit o Fiur 6. No implis n, whil n throuh n nin no (^) imply, so trnsitiv losur rph hs n rom no to no. Similrly, w t n rom no to no. Th ontrpositiv s or ths two s r: implyin n implyin, rsptivly. Now, rom no nothr nin no oms tru (with no, implyin ). So, w t, nothr trnsitiv losur rom to n its ontrpositiv rom no to no. Also, w t trnsitiv losur rom to. Now, i w strt our trvrsl rom no, w t implition to n implis. Thus, w t n trnsitiv losur rom to (shown ol in Fiur 7), whih intis s--0 ult on lin to runnt. Th rsults o Stion 6 (Tl 1) os not ontin th implmnttion o th ontrpositiv rul. 5 Limittions Th nout stm in Fiur 8 is unosrvl n th omintor sts or stm r nithr x nor unosrvl. Thus, Thorms 3.1 n 3.2 il to intiy th unosrvility sttus o stm. Fiur 9 shows iruit with six runnt ults s oun y natpg prorm. Th nout stm oul not lssi s unosrvl y our thorms us no x-vlu or unosrvl omintor sts r oun. So, th runnt ults s--1 n s--0 oul not inti runnt y our thniqu. Fults s--1, 1 s--1, n s--1 r inti s runnt. Th ult s--1 is not inti us w o not hv nouh implitions to trvrs kwrs rom th tru Fiur 7: A prt o trnsitiv losur with ontrpositiv s to n runnt s--0 ult in Fiur Fiur 8: Exmpl o n unosrvl stm not inti y stm unosrvility thorms. no o th output o NAND t towrs its inputs, i.., w o not hv implitions to trvrs kwrs rom th tru nos o sinl n o Fiur 9. Du to this, w n not t n implition to intiy th unrivl runnt ult s--1. An xtn implition mtho in whih sts o sinls r numrt to hk or ssntil ssinmnts n intiy suh runny t hihr omputtion ost [21]. Th thr runnt ults not inti in th iruit 432 r o this typ (rr to Tl 1). 6 Rsults Tl 1 shows th rsults otin on ISCAS '85 n ISCAS '89 nhmrk iruits. Th rst olumn shows th nms o nhmrk iruits or whih rsults rom vrious prorms r ompr with rspt to th runnt ults inti n thir rsptiv CPU tims. Th nxt olumn shows th totl numr o ollps sinl ults in th iruit. Th nxt two olumns show th runnt ults inti n CPU tim in sons or th ATPG tool TRAN [4] y Chkrhr t l., whih uss trnsitiv losur or tst nrtion. Th nxt two olumns lists th rsult o FIRE [13] yiyr n Armovii. Th nxt two olumns show th rsults o th trnsitiv losur lorithm without prtil implitions y Arwl t l. [1]. Th nxt two olumns show th rsult o th trnsitiv losur lorithm with som o th prtil implitions y Gur t l. [7]. Th lst two olumns r th rsults o th nw ult-inpnnt lorithm, whih uss trnsitiv losur or runny intition. Th squntil nhmrk iruits wr onvrt into omintionl iruits y rmovin th ip-ops rom th iruit n trtin ip-op outputs s primry inputs n ip-op inputs s primry outputs. Prpr or DATE'04 P4o6

5 Tl 1: Comintionlly runnt ults in ISCAS '85 n ISCAS '89 nhmrk iruits. Ciruit Totl Runnt ults inti n run tim on Sun worksttions (: Spr 5; : Spr 2) ults TRAN [4] FIRE [13] TC [1] TC w prtil [7] OurAlo: Run. CPU Run. CPU Run. CPU Run. CPU Run. CPU ults s () ults s () ults s () ults s () ults s () s s s s s s s s s Fiur 9: Anothr xmpl o limittions. Thr r no ort ults in TRAN [4] or ll iruits in Tl 1. W us nhmrk iruit 5315 to ompr rsults or ll ths irnt prorms with our work. Th ATPG tool TRAN [4] intis ll th 59 runnt ults in th iruit without ortin ny ult in CPU tim o 32.3 s. FIRE [13] intis 20 runnt ults with CPU tim o 2.8 s. Nxt, th trnsitiv losur lorithm without prtil implition [1] intis 20 runnt ults with CPU tim o 3.4 s. Nxt, th trnsitiv losur lorithm with som prtil implitions [7] intis 32 runnt ults with CPU tim o 3.4 s. Finlly, our lorithm with th x-vlu thorm n stm unosrvility thorms implmnt intis 58 runnt ults in 3.9 s. Thus, in th nhmrk iruit 5315 our lorithm intis mor runnt ults thn ny o th ult inpnnt lorithms. Th runnt ults inti y our work r muh rtr thn ll th othr implition-s mthos sri in th rsult tl with omprl CPU tim. Th rsults r lso omprl to th ATPG tool h TRAN us or th nhmrk iruits 3540, 5315, 6288, s349, s444, s713, s1423 n s9234 our work provis ithr ll or los to ll runnt ults. W o roniz tht thr r ATPG prorms [28] tht r str thn TRAN, ut thn our prsnt implmnttion is only xprimntl with mny potntil prormn improvmnts possil. An nlysis o th nhmrk iruit 432 rvls tht u to th inility to tk ision (i.., trvrs kwrs rom th ls no o th output o th AND t, shown in Stion 2) w n not intiy 3 out o 4 runnt ults (i.., w intiy only 1 out o 4 runnt ults.) 7 Conlusion Th x-vlu thorm n stm unosrvility thorms nhn th trnsitiv losur thniqu to intiy runnt ults. Its othr pplitions, not isuss hr ut unr invstition, inlu th intition o th quivlnt ults n quivln hkin o two ivn iruits. Futur rsrh my n nw wys o tkin isions in th implition rph to trvrs rom th ls sttus o th output sinl o n AND t. Howvr, our thniqu rtins lowr omplxity thn th xponntil omplxityonatpg lorithm n intis mny, thouh not ll, runnt ults. Rrns [1] V. D. Arwl, M. L. Bushnll, n Q. Lin, \Runny Intition Usin Trnsitiv Closur," in Pro. Prpr or DATE'04 P5o6

6 o th 5th Asin Tst Symp., Novmr 1996, pp. 4{9. [2] M. L. Bushnll n V. D. Arwl, Essntils o Eltroni Tstin or Diitl, Mmory n Mix-Sinl VLSI Ciruits. Boston, MA: Kluwr Ami Pulitions, [3] S. T. Chkrhr, V. D. Arwl, n M. L. Bushnll, \Nurl Nt n Booln Stisility Mols o Loi Ciruits," IEEE Dsin n Tst o Computrs, vol. 7, no. 5, pp. 54{57, Otor [4] S. T. Chkrhr, V. D. Arwl, n S. G. Rothwilr, \A Trnsitiv Closur Alorithm or Tst Gnrtion," IEEE Trns. on Computr-Ai Dsin, vol. 12, no. 7, pp. 15{1028, July [5] H. Fujiwr n T. Shimono, \On th Alrtion o Tst Gnrtion Alorithms," in Pro. o th Intrntionl Fult-Tolrnt Computin Symp., Jun 1983, pp. 98{105. [6] V. Gur, \A Nw Trnsitiv Closur Alorithm to Intiy Runnis in Loi Ciruits," Mstr's thsis, Rutrs Univrsity, Jnury [7] V. Gur, V. D. Arwl, n M. L. Bushnll, \A Nw Trnsitiv Closur Alorithm with Applitions to Runny Intition," in Pro. o th 1st Intrntionl Workshop on Eltroni, Dsin n Tst Applitions (DELTA'02), Jnury 2002, pp. 496{500. [8] P. Gol, \An Impliit Enumrtion Alorithm to Gnrt Tsts or Comintionl Loi Ciruits," IEEE Trns. on Computrs, vol. C-30, no. 3, pp. 215{222, Mrh [9] L. H. Golstin, \Controllility/Osrvility Anlysis o Diitl Ciruits," IEEE Trns. on Ciruits n Systms, vol. CAS-26, no. 9, pp. 685{693, Sptmr [10] F. Hrry, Grph Thory. Rin, MA: Aison- Wsly, [11] M. Hrihr n P. R. Mnon, \Intition o Unttl Fults in Comintionl Ciruits," in Pro. o th Intrntionl Con. on Computr Dsin, Otor 1989, pp. 290{293. [12] M. Hntlin, H. Wittmnn, n K. J. Antrih, \A Forml Non-Huristi ATPG Approh," in Pro. o th Europn Dsin Automtion Con., Sptmr 1995, pp. 248{253. [13] M. A. Iyr n M. Armovii, \FIRE: A Fult- Inpnnt Comintionl Runny Intition Alorithm," IEEE Trnstions on VLSI Systms, vol. 4, no. 2, pp. 295{3, Jun [14] T. Kirkln n M. R. Mrr, \A Topoloil Srh Alorithm or ATPG," in Pro. o th 24 th Dsin Automtion Con., Jun-July 1987, pp. 502{508. [15] W. Kunz n D. K. Prhn, \Rursiv Lrnin: An Attrtiv Altrntiv to th Dision Tr or Tst Gnrtion in Diitl Ciruits," in Pro. o th IEEE Intrntionl Tst Con., Sptmr 1992, pp. 816{825. [16] T. Lrr, \Tst Pttrn Gnrtion Usin Booln Stisility," IEEE Trnstions on Computr-Ai Dsin, vol. 11, no. 1, pp. 4{15, Jnury [17] V. Mht, \Runny Intition in Loi Ciruits usin Extn Implition Grph n Stm Unosrvility Thorms," Mstr's thsis, Rutrs Univrsity, My [18] V. J. Mht, K. K. Dv, V. D. Arwl, n M. L. Bushnll, \A Fult-Inpnnt Trnsitiv Closur Alorithm or Runny Intition," in Pro. o th 16 th Intrntionl Con. VLSI Dsin, Jnury 2003, pp. 149{154. [19] P. R. Mnon n H. Ahuj, \Runny Rmovl n Simplition o Comintionl Ciruits," in Pro. o th 10 th IEEE VLSI Tst Symp., April 1992, pp. 268{ 273. [20] K. P. Prkr n E. J. MClusky, \Proilisti Trtmnt o Gnrl Comintionl Ntworks," IEEE Trns. on Computrs, vol. C-24, no. 6, pp. 668{670, Jun [21] Q. Pn, M. Armovii, n J. Svir, \MUST: Multipl-stm Anlysis or Intiyin Squntilly Untstl Fults," in in Pro. o th Intrntionl Tst Con., Sptmr 2000, pp. 839{846. [22] I. M. Rtiu, A. Sniovnni-Vinntlli, n D. O. Prson, \VICTOR: A Fst VLSI Tstility Anlysis Prorm," in Pro. o th IEEE Intrntionl Tst Conrn, Novmr 1982, pp. 397{4. [23] J. P. Roth, \Dionosis o Automt Filurs: A Clulus n Mtho," IBM Journl o Rsrh n Dvlopmnt, vol. 10, no. 4, pp. 278{291, July [24] M. H. Shulz n E. Auth, \Avn Automti Tst Pttrn Gnrtion n Runny Intition Thniqus," in Pro. o th Intrntionl Fult- Tolrnt Computin Symp., Jun 1988, pp. 30{35. [25] M. H. Shulz, E. Trishlr, n T. M. Srrt, \SOCRATES: A Hihly Eint Automti Tst Pttrn Gnrtion Systm," IEEE Trns. on Computr- Ai Dsin, vol. CAD-7, no. 1, pp. 126{137, Jnury [26] S. C. Sth n V. D. Arwl, \A Nw Mol or Computtion o Proilisti Tstility in Comintionl Ciruits," Intrtion, th VLSI Journl, vol. 7, no. 1, pp. 49{75, April [27] J. P. M. Silv n K. A. Skllh, \Grsp - A nw Srh Alorithm or Stisility," in Pro. o th Intrntionl Con. on Computr-Ai Dsin, Novmr 1996, pp. 220{227. [28] P. Stphn, R. K. Bryton, n A. L. Sniovnni- Vinntlli, \Comintionl Tst Gnrtion Usin Stisility," IEEE Trns. on Computr-Ai Dsin, vol. 15, no. 9, pp. 1167{1176, Sptmr [29] J. K. Zho, E. M. Runik, n J. H. Ptl, \Stti Loi Implition with Applition to Runny Intition," in Pro. o th 15 th IEEE VLSI Tst Symp., April 1997, pp. 288{293. Prpr or DATE'04 P6o6