35 th Design Automation Conference Copyright 1998 ACM

Transcription

1 Eint Booln ivision n Sustitution Shih-Chih Chng vi Ihsin Chng pt. o CS n IE Ntionl Chung Chng Univrsit Ultim Intronnt Th. Chi-Yi, TAIWAN, ROC Sunnvl, CA Astrt Booln ivision, n hn Booln sustitution, prous ttr rsult thn lgri ivision n sustitution. Howvr, u to th lk o n int Booln ivision lgorithm, Booln sustitution hs rrl n us. W prsnt n int Booln ivision n sustitution lgorithm. Our thniqu is s on th philosoph o runn ition n rmovl. B ing multipl wirs/gts in spiliz w, w tilor th philosoph onto th Booln ivision n sustitution prolm. rom th viwpoint o tritionl ivision/sustitution, our lgorithm n prorm sustitution not onl in sum-o-prout orm or ut lso in prout-o-sum orm. Our lgorithm n lso nturll tk ll tps o on't rs into onsirtion. As r s sustitution is onrn, w lso isuss th s whr w r llow to ompos not onl th ivin ut lso th ivisor. Exprimnts r prsnt n th rsult is promising. 1 Introution In multi-lvl logi snthsis, n importnt stp in minimizing th r o iruit is sustitution [4] (or rsustitution [10]). Sustitution rrs to th stp whr untion is simpli in omplxit using n itionl input tht ws not prviousl in th untion's immit nins. Sustitution n ru th omplxit o untion us prt o th untion is rpl th itionl input tht rprsnts som xisting untion in th iruit. Th xprssion o th xisting untion is thror shr n rus. To prorm sustitution, th onpt o ivision pls mjor rol. Givn two Booln untions n, i w n xprss in th orm = Q + R, whr n + rsptivl rprsnt th Booln AN n Booln OR oprtors, thn w s tht n ivi n tht untions Q n R r, rsptivl, th quotint n th rminr. Sustitution n lgri or Booln, pning on i th unrlining ivision is lgri or Booln. In lgri ivision [4], logi xprssions r trt s lgri polnomils, with som rstritions pl on th mnipultions o th polnomils. In prtiulr, th prout o two untions G is lgri onl i no vril pprs in oth n G. As onsqun o th rstrition, rtin x-98/0006/$ th sign Automtion Conrn Copright 1998 ACM Booln intitis suh s xx = 0 n xx = x o not xist. As n xmpl, givn = + + n ivisor = +, through lgri ivision w otin = ( + )+. Through Booln ivision, whih n xploit ll th proprtis in Booln lgr [1], w otin = ( + )( + ). Assuming no with untion + xists in th iruit, with lgri sustitution w thn hv = +, whil with Booln sustitution w hv=(+ ). In this xmpl, untion hs 5 litrls 1 or sustitution. Algri sustitution rus th numr o litrls to 4, whil Booln sustitution rus it to 3. Booln ivision, n hn Booln sustitution, in thor prous ttr rsults. Howvr, thr os not xist gnrl n int Booln ivision lgorithm. In trms o th ov xmpl, this mns tht th st rsult o ruing to 3 litrls is vr iult to hiv. In this ppr, w rst prsnt nw thniqu to prorm Booln ivision. Our thniqu is s on th onpt o runn ition n rmovl (RAR) isuss in [2][3][5][8]. Th si philosoph o th RAR thniqu is to rst som runn n thn rmov othr runnis lswhr, with th gol tht th rmov ons ru th iruit siz mor thn th on. With x stup tht is spill ongur, w tilor th RAR philosoph onto th Booln ivision prolm. Unlik tritionl RAR thniqus, whih rquir runn hking on th potntil wir to, our lgorithm is tilor in w tht w know priori tht our intrst potntil wir is runnt. Also, lthough quit tiv on ing on runn n thn rmoving othr runnis, th tritionl RAR thniqus hv littl suss on tring to multipl wirs/gts. In our lgorithm, th tritionl RAR philosoph istilor to multipl wirs/gts in spi w prtiulrl or th Booln ivision prolm. As r s sustitution is onrn, knowing how to prorm ivision is onl th rst stp. Th son stp is to hoos potntil ivisors. Tritionll, sustitution on untion is on going through th xisting nos in th iruit n trting h o thm 1 In tor orm [4], s oppos to sum-o-prout orm. AC98-06/98 Sn rniso, CA USA

2 s potntil ivisor o. ivision is tri on h potntil ivisor n sustitution is rri out whn th tril is vorl. Sin it is up to th unrlining ivision lgorithm to onlu i ivisor is goo or not, th lgorithm m miss som \goo" ivisors. In th xmpl mntion rlir, lt us s th no with untion + os not xist n, inst, no with untion = + + x xists. Sin untion os not pn on vril x, tritionl ivision lgorithm woul quikl onlu tht th quotint o untion ivi is 0, n thror no sustitution woul our. Howvr, i w slightl hng th iruit strutur omposing + + x to two nos n 1 = + n n 2 = n 1 +x, untion n thn sustitut with no n 1. W will us th trm si ivision to rr to th snrio whr th givn ivisor is not llow to ompos, n th trm xtn ivision or th snrio whr th ivisor is llow to ompos, rtinl with som purpos in min. In th ov xmpl whr untion = ++ is ivi = ++x, wwoul s tht unr si ivision th quotintis0. or th sm n w woul lso s tht unr xtn ivision th suxprssion + n xtrt out s nw ivisor, n with th nw ivisor + th quotint is+. rom this viwpoint, ll th tritionl ivision lgorithms prorm onl si ivision, whil our lgorithm prsnt in this ppr prorms xtn ivision. Tritionl sustitution pprohs oprt on h no's intrnl sum-o-prout t strutur, n hn n onl prorm sustitution/ivision in th sum-o-prout orm. In ontrst, our lgorithm oprts on iruit strutur irtl. Givn n initil iruit, th rst stp o our lgorithm is to ompos h no's intrnl sum-o-prout orm into two-lvl AN n OR gts. Th iruit thn, in gnrl, hs lvl o AN gts, ollow lvl o OR gts, n so on. As rsult, in ition to th tritionl sum-o-prout tp o sustitution, our lgorithm n lso prorm sustitution in th vor o prout-o-sum orm. In othr wors, in two-lvl orm, whthr th ivin/ivisor r unh o ANs ollow n OR, or unh o ORs ollow n AN r ompltl smmtri to us. or xmpl, lt = ( + )( + )( + ) n = ()() xisting nos. With our lgorithm w n quikl sustitut into n otin = +, i.., = +. Prorming sustitution in suh mnnr is ompltl not possil in th tritionl pprohs us o th strong tthmnt to th unrlining sum-o-prout xprssion, whil in our thniqu prorming sustitution through sumo-prout orm or prout-o-sum orm r sill th sm. 2 Runn ition n rmovl g1 g2 g3 g4 g5 g6 g7 () g8 g9 o1 o2 Th most rlt work to our Booln sustitution lgorithm is th thniqu o runn ition n rmovl (RAR). Hr w provi til rviw. In [2][3][5][8], th thniqu o RAR is propos n ppli to gnrl multi-lvl logi optimiztion. Th g1 g2 g3 g5 () igur 1: Th RAR thniqu si philosoph in RAR is to som runn rst n thn tr to rmov othr runnis lswhr, with th gol tht th rmov ons ru th iruit siz mor thn th on. W rviw th thniqu with n xmpl iruit. ig. 1(), without th ott wir, shows n irrunnt iruit. Th ott wir g5!g9 is runnt wir, i.., ing th wir os not hng th iruit's untionlit. Howvr, on this wir is, th two thik wirs, g1!g4 n g6!g7, om runnt. In this s, w n rmov ths two runnt wirs without hnging th iruit's untionlit. Atr rmoving ths two wirs, w thn hv th iruit shown in ig. 1(), whih is smllr in siz. In gnrl, th RAR thniqu rst is, s on som ost untion, som xisting irrunnt wir tht is th trgt to rmov. Thn th thniqu srhs or som non-xisting wir, somtims ll nit onntion, tht on n rmov th trgt wir. inll th thniqu hks i th nit onntion is runnt, i.., i ing th non-xisting wir prsrvs th iruit's untionlit. Onl whn th nit onntion is vri s runnt, w n thn th onntion n urthr rmov th trgt wir. Not tht most o th RAR thniqus onl tr to inrmntll on wir t tim. u to high srh sp, orts tht tr to multipl wirs/gts n rmov vn mor wirs/gts hv onl littl suss (.g.: [2]). 3 Bsi ivision Givn untion n ivisor, w us th trm si ivision to rr to th snrio whr th ivisor is not llow to ompos, n us th trm xtn ivision to rr to th snrio whr th ivisor n rl ompos, with som optimiztion gol in min. In this stion w ous on si ivision. 3.1 SOS n POS o untion W rst n som nitions. A prout trm, or u, is st o litrls AN' togthr. A sum trm is st o litrls OR' togthr. A untion 1 ontins untion 2 i th on-st o 1 ontins th on-st o 2. As n xmpl, untion (u) ontins untion (u) ; untion (sum trm) + ontins untion (sum trm). urthrmor, w n SOS n POS o untion s ollows: SOS: Givn untion in two-lvl sum-oprout orm, w s untion G, lso in sumo-prout orm, is sum-o-suprout, or SOS, g8 g9 o1 o2

3 o i vr u in is ontin t lst on u in G. POS: Givn untion in two-lvl prout-osum orm, w s untion G, lso in prouto-sum orm, is prout-o-susum, or POS, o i vr sum trm in ontins t lst on sum trm in G. or xmpl, = + is SOS o = ++ us vr u in is ontin ithr u or u in. or nothr xmpl, 0 = + + x is lso SOS o th ov, sin ing mor us to os not hng th originl ontinmnt rltionship in. On th othr hn, untion E = + is not SOS o, sin u is not ontin in n u in untion E. On th POS si, or xmpl, =()() ispos o =(+)( + )( + ) us vr sum trm in ontins ithr sum trm or sum trm in. or nothr xmpl, untion 0 =()()(x + ) is lso POS o th ov, sin ing mor sum trms to os not hng th originl ontinmnt rltionship in. On th othr hn, untion E = ()(+) is not POS o, sin sum trm + os not ontin n sum trm in untion E. Th onpts o SOS n POS pl ntrl rol in our lgorithm, n w now look t som o thir simpl proprtis. Lmm 1 Lt untion G SOS o untion. Thn = G. Lmm 2 Lt untion G POS o untion. Thn = + G. Ths two lmms stlish th groun whr w n tilor th thniqu o runn ition n rmovl (RAR) onto our sustitution prolm. To illustrt th onpt, w tk th xmpl o = + + n = + rom Stion 1. Sin is SOS o, Lmm 1, th nw untion nw = ( + )( + + ) must quivlnt to th originl untion. rom th RAR viwpoint, w hv sussull \" runn into th iruit. ousing on th originl prt insi nw,w thn tr to rmov s mn runnis s possil, n n quikl rriv =( +)( + ). Smmtri to th SOS s, w n prorm similr oprtions on POS. Lt =(+)( + )( + ) n =()(). Sin is POS o, Lmm 2, th nw untion nw =()()+(+)( + )( + ) must quivlnt to th originl untion. ousing on rmoving runnis rom th originl prt insi nw,w thn quikl hv = Prorming si ivision Givn untion n ivisor, in this stion w prsnt n lgorithm tht prorms si Booln ivision, i.., = Q + R. Th st w to xplin our lgorithm is to isuss it with n xmpl. ig. 2() shows two nos, whih orrspon to = n = +. Sin our ntrl i is s on th SOS onpt, th rst stp to () () Q () R \R R () () igur 2: Bsi ivision 0 0 g 1 g 2 g 3 g 4 g 5 g 6 Q init 1 Q init prorm ivi is to tk out rom ll th us tht r not ontin n u in, n suh us will our nl rminr trm R. Among th our us in, is th onl suh u sin 6 n 6. ig. 2() shows th iruit strutur tr w orm th rminr, whr w us ott irl R to init th rminr rgion n \R to not th rsulting untion with u tkn out rom. Sin vr u in \R is now ontin t lst on u in, is SOS o \R. B Lmm 1, \R woul st unhng i AN' with. This t is shown in ig. 2() with n xtr ol AN gt n th shit o \R rom or this AN gt to tr this AN gt. rom th viwpoint o th RAR thniqu, w hv sussull runn n th iruit still hs th sm untionlit. Now th rgion mrk th irl Q init is highl runnt. Th nl stp is to prorm runn rmovl on th Q init rgion n w rh th nl rsult shown in ig. 2(), whih is o th orm = Q + R. To show howrunn rmovl is on, w uplit th iruit snpshot shown in ig. 2() to ig. 2() n rmrk som nos. Lt us illustrt how wir!g 2, th thik wir in ig. 2(), is tt s runnt wir. or wir!g 2 stuk-t-1 ult to tstl, must 0 to tivt th ult. or th ult t to propgt through gt g 2, must 0. or th ult t to propgt through gt g 6, g 1 must 1. Sin = 0 n = 0 implis + = 0, gt g 1 must 0, whih is onit. A onit uring th implition pross mns th ult!g 2 stuk-t-1 is untstl, n thror wir!g 2 n rpl onstnt 1. Our si ivision lgorithm works s illustrt th ov xmpl. In summr, our lgorithm onsists o thr stps. Th rst stp o our lgorithm is to ompos th ivin so tht th us tht mk th ivisor not SOS o orm th rminr R. Th son stp is to AN with \R, whih os not hng th untionlit o\r Lmm 1. Th thir stp is to rmov ll th r- \R R

4 unnis insi th \R rgion. Not tht it is th runn ition n rmovl stps tht mk our thniqu Booln. Compring to th tritionl RAR thniqus, howvr, mjor irn lis on th t tht w know priori tht th wirs/gts r runnt us o th SOS proprt in Lmm 1. In othr wors, unlik th tritionl RAR thniqus, w o not n to hk i th wir/gt r runnt or not. urthrmor, s mntion in Stion 2, thr hs n littl suss in works tring to gnrliz th RAR thniqu to ing multipl wirs/gts. Wht our lgorithm os is ssntill tilor vrsion o th RAR philosoph onto th sustitution prolm, with x ongurtion o multipl wirs/gts ition. Also not tht sin th wirs/gts r known to runnt priori, th most tim-onsuming stp in our lgorithm is onl on th runn rmovl stp. With irnt implition mthos (.g.:[6][7][9]) w n tull just th tro twn th run tim n th qulit o rsult. or xmpl, w n limit our implition pross onl insi smll rgion, th \R rgion plus th rgion. As r s sustitution is onrn, most o th ronvrgns n implition onits woul our in this smll rgion. Limiting th implition pross insi this smll rgion woul grtl ru th tim rquir s oppos to tritionl runn rmovl pross. On th othr hn, w n rtinl spn mor tim to prorm implitions to gts outsi this smll rgion, n thr nturll inorporting n xtrnl or intrnl on't rs into onsirtion. In th xtrm s, w n vn opt som quit xhustiv implition thniqu suh s rursiv lrning [7] to inorport lrg mount o intrnl on't rs. W o not isuss th tils hr ut simpl point out th xistn o suh xiilit on vrious implition lgorithms. inll, s n sn rom th ov xmpl, our lgorithm oprts on iruit strutur irtl, rthr thn mnipulting xprssions lik tritionl pprohs. As mntion rlir, w thror r not limit to oing sustitution onl in trms o th tritionl sum-o-prout viwpoint. With th POS onpt, w n lso prorm sustitution on two untions whn th r oth in th prout-o-sum orm. Inst o using th SOS onpt n Lmm 1, w n us th POS onpt n Lmm 2, n th sm philosoph s illustrt ov woul ppl irtl. As simpl xmpl, imgin iruit tht is intil to th on shown in ig. 2 with ll th AN gts hng to OR gts n vi vrs. With our lgorithm it is s s s ws illustrt in this stion, whil in tritionl sustitution thniqu ll th sum-o-prout xprssions orm omplt nw prolm whos rsult is iult to prit. Sin onptull SOS n POS r smmtri, throughout th rmining o this ppr w o not go into th tils o th s or POS. 4 Extn ivision Th prvious stion prsnt our lgorithm tht prorms si ivision, whr ivisor is not llow to ompos. Givn untion n ivisor, unr si ivision w sk to rxprss s = Q + R. This mns w r onl llow to ompos ut not on. In this stion w prsnt n lgorithm tht prorms wht w ll xtn ivision. Givn untion n ivisor, unr xtn ivision w r llow to ompos not onl ut lso, with th purpos o minimizing th numr o litrls in sustitution. In ssn, w rst wnt to sprt th us in into two groups, th or ivisor C n th rmining ivisor R. On this sprtion is trmin, w ompos th originl ivisor into two nos suh tht = C + R. omposing into nw no or th or ivisor C mns tht C, suxprssion tht ws originll m in th givn ivisor, isnow xpos n n us or sustitution. W thn ppl our si ivision lgorithm in th prvious stion on untion n or ivisor C to otin th rsult. or xmpl, givn untion = n ivisor = + + x, w ompos th ivisor into th or ivisor C = + n th rmining ivisor R = x. Appling our si ivision lgorithm on n C, w thn otin th sm rsult s illustrt in th prvious stion. It shoul lr tht th most importnt thing hr is to intlligntl trmin th or ivisor C, sin on C is trmin n xtn ivision rus to si ivision. Rll tht uring our si ivision lgorithm, it is th stp o runn rmovl tht rll prorms th minimiztion pross. Looking k in ig. 2(), whnvr w rmov wir rom th us in th Q init rgion, w tivl ru litrl in th nl quotint. Wht w woul lik to hv is or ivisor tht is l to rmov th most wirs. To trmin th or ivisor C with givn untion n givn ivisor, our si i is to hv h wir in th us o \vot" or nit or ivisor. or h wir w in th us o,w prorm implitions to s whih us in ivisor r l to rmov wir w. or xmpl, lt untion = n ivisor = , whos iruit strutur is shown in ig. 3(). In ig. 3(), w nm ivisor 's v us 1 ; 2 ; 3 ; 4 n 5 ; w lso nm untion 's our us x 1 ;x 2 ;x 3 n x 4, whih r rsptivl rivn gts g 1 ;g 2 ;g 3 n g 4. Consir wir!g 1 stuk-t-1 ult. W hv th ollowing implitions. = 0 (tivt ult) =) 1 = 0 =1&= 1 (llow ult thru g1) =) 4 =1 x2= 0 (llow ult thru g5) & =0 =) =0 =0 =) 2=0 Assum or now tht w somhow hv trmin or ivisor C. This or ivisor, in our spiliz ongurtion or si ivision, s into gt similr to th ol AN gt g 6 in ig. 2() o th prvious stion. This mns tht i w wnt n ult t in th Q init rgion to propgt through th ol AN gt, this or ivisor C must hv vlu 1 uring th ult's implition pross. In th s o xtn ivision, i th or ivisor tht w vntull tr-

5 () x 1 g 1 g 2 x 2 x 3 g 3 x 4 g 4 g 5 2=0 1 =0 () Q init igur 3: Extn ivision min hs implition vlu 0 or prtiulr ult, th ult must untstl us onit will our with th rquir ssignmnt o 1 mntion ov. W illustrt this point ontinuing th xmpl or wir!g 1 stuk-t-1 ult. W ous on th rsults tht ppr on th i 's si whos implition vlus r 0. In this s, w hv 1 = 0 n 2 =0. Assuming w vntull hoos s our nl or ivisor, i.., C = = +, thn our si ivision lgorithm in th prvious stion woul hng th iruit strutur to th on shown in ig. 3(), whr C = + is onnt to th ol AN gt. ollowing th si ivision lgorithm in th prvious stion, w woul tr to rmov smn wirs s possil in th Q init rgion. Whn w gin prorm implitions or th ult!g 1 stuk-t-1 ult, shown with ross in ig. 3(), w know tht 1 n 2, n hn C, ll hv implition vlu 0. This rts onit us, s stt rlir, or th ult t o!g 1 stuk-t-1 to propgt through th ol AN gt, C must ssign 1. In othr wors, i w ohoos s our or ivisor, w xpt wir!g 1 to rmov in th susqunt si ivision. Now, in trmining th or ivisor, irnt wirs hv irnt implition vlus on th i 's si in ig. 3(). In som sns, this mns tht h wir \vots" or nit or ivisor. In th ov xmpl, wir!g 1 vots or nit or ivisor This shoul om lr i w look t th omplt sitution tr h wir prorms implitions on th xmpl iruit shown in ig. 3(). Tl 1() lists ll th i 's tht hv implition vlu 0 or h wir. W xplin th intrprttion o Tl 1() xmpls. Th mning o th son row is tht w xpt wir! g 1 to rmov i w hoos s th or ivisor. or simpliit, w lso s tht wir!g 1 vots or nit or ivisor Similrl, th mning o th ourth row is tht w o not xpt wir!g 2 to rmov, rgrlss o whtvr or ivisor w () 1 2 wir i =0!g1!g1!g1!g2!g2!g3!g3!g4!g4 ;4;5 4;5 4;5 1 4;5 wir i =0!g1!g1!g1!g2!g2!g3!g3!g4!g4 ;4;5 4;5 4;5 () initil () nl Tl 1: Vot tl hoos, n hn hs no nit or ivisor. Th rmining ntris o Tl 1() n intrprt in similr w. Th ov voting shm monstrts our ritri or hoosing goo or ivisor. rom th RAR thniqu's viwpoint, howvr, on mor thing w n to mk sur is tht nit or ivisor is in runnt wir whih w n vntull \" to th iruit. This is on hking i th nit or ivisor vot wir w is SOS o th u tht is onnt to wir w. or xmpl, rom th rst ntr in Tl 1(), th nit or ivisor o wir!g 1 is = +. Th u tht is onnt to wir!g 1 is x 1 =. Sin th nit or ivisor + is SOS o u, w know vntull i w or ivisor + into th iruit, th wir will runnt wir n thror th iruit untionlit woul not hng. In Tl 1(), th onl nit or ivisors tht o not hol or this onition r wirs!g 3 n!g 4. Th nit or ivisor or wir!g 3 is = +, whih is not SOS o th orrsponing u x 3 =. On th s o wir!g 4, nit or ivisor 1 = is not SOS o th orrsponing u x 4 =. W thror n to lt ths two ntris in Tl 1() n w hv our nl vot tl, shown in Tl 1(). To nliz th hoi o th or ivisor, vrious huristis n us. W ru th ov hoi prolm to mximl liqu prolm in grph thor. u to sp limit, w omit th tils hr. Appling our xtn ivision lgorithm to th sustitution prolm, w wnt to point out tht w n tull o mor thn wht th ov isussion shows. In th ov ormultion, w ous onl on on xisting no. In th s o sustitution, w tull hv rom to slt our or ivisor rom mong mn iruit nos. As n xmpl o how this gnrliztion works, imgin th givn ivisor in th ov xmpl, = , os not xist in our iruit n inst, two nos 1 = + + n 2 = + xist, s shown in ig. 3(). Whn untion = is givn n w wnt to srh or goo ivisor twn 1 n 2 with xtn ivision, w n tmporril prtn tht ll th v us r rom th sm no, n thror th ow is intil to th xmpl shown in this stion. Eh wir in th us o 1 n 2 vots or nit or ivisor n w hvnintil vot

6 nh init sis si xt. xt.+gc lit. lit. pu lit. pu lit. pu lit. pu C C C C C C C C lu px lu s rg i i i rot t trm x x s s s s s s totl % 1.2% 9.5% 9.8% 10.5% Tl 2: Exprimntl rsults tl s shown in Tl 1(). Th onl slight moition w n is in th nl mximl liqu ormultion, whr w n to mol th t tht som us in th son olumn o Tl 1() originll om rom irnt no. Not tht, s is lso th s or si ivision, w n prorm xtn ivision in trms o sum-o-prout orm s wll s prout-o-sum orm. Inst o ousing on th us tht hv implition vlu 0, w woul thn ous on th sum trms tht hv implition vlu 1. Th rst o th lgorithm pplis similrl. 5 Exprimntl rsults W hv implmnt our lgorithm n ppli it to th sustitution prolm. Our implmnttion hs thr ongurtions: si ivision, xtn ivision without onsiring glol intrnl on't rs, n xtn ivision with glol intrnl on't rs tkn into onsirtion. W prorm xprimnts on MCNC n ISCAS nhmrks within SIS [10] nvironmnt. W rst run th sript limint 0; simpli; gx; on h nhmrk to otin th initil iruit. W thn ompr our lgorithm with th lgri rsustitution \rsu -" in SIS. Tl 2 shows th omprison twn SIS n our rsults. Th rst olumn shows th nm o th iruit. Th son olumn shows th initil litrl ount. Th olumns ll \sis" is th rsult o running \rsu -" ommn, with suolumns \lit." n \pu" rporting th numr o litrls n CPU tim, rsptivl. All litrl ounts r in tor orm. Tk th iruit C2670 s n xmpl, tr running th ov sript, initill th iruit h 939 litrls, shown in th son olumn. Atr running \rsu -" th litrl ountws ru to 849 litrls. With our si ivision it is ru to 840 (Column \si"). Th xtn ivision ru it to 831 (Column \xt."). Th xtn ivision with glol on't rs tkn into ount rought itown to 828 (Column \xt.+gc"). Th lst two rows show th summtion o h olumn n th prntg o improvmnt ompr to th initil litrl ount. As th tl inits, our ivision lgorithms outprorm th tritionl ivision n sustitution. In trms o run tim, \rsu -", th si ivision, n th xtn ivision without glol on't rs spnt similr CPU tims. Th xtn ivision with glol on't rs spnt th most tim. 6 Conlusion In this ppr w rst prsnt n int lgorithm, s on th philosoph o runn ition n rmovl (RAR), or prorming Booln ivision. With th onpt o SOS n POS, w tilor th RAR philosoph to th Booln ivision prolm. Th tiloring nls us to multipl wirs/gts in spiliz ongurtion n rmov mor wirs/gts. Appling our Booln ivision lgorithm, our lgorithm n prorm sustitution not onl in th tritionl sum-o-prout orm, ut lso in prout-o-sum orm. W thn gnrliz our si ivision to wht w ll xtn ivision. Extn ivision llows us to ompos not onl on th ivin ut lso on th ivisor. urthrmor, our thniqu is l to nturll inorport ll tps o on't rs into onsirtion. W lso prsnt som xprimntl rsults to vri th tivnss o our lgorithm. Rrns [1] R.K. Brton n C. MMulln, \Th omposition n toriztion o Booln Exprssions," Pro. ISCAS, pp , [2] S.C.Chng n M.Mrk-Sowsk, \Prtur n simpli: multi-lvl Booln ntwork optimizr," Pro. ICCA, pp. 2-6, Nov [3] S.C. Chng, L. VnGinnkn, n M. Mrk-Sowsk, \st Booln Optimiztion Rwiring," Pro. ICCA, pp , [4] G. Mihli, \Snthsis n Optimiztion o igitl Ciruits," M Grw Hill Txt, [5] L.A.Entrn n K.T.Chng, \Comintion n squntil logi optimiztion runn ition n rmovl," IEEE TCA, Vol. 14, no. 7, pp , Jul [6] T. Kirkn n M.R. Mrr, \A Topologil Srh Algorithm or ATPG," Pro. AC, pp , [7] W. Kunz n.k. Prhn, \Rursiv Lrning: An Attrtiv Altrntiv to th ision Tr or Tst Gnrtion in igitl Ciruits," Pro. ITC, pp , [8] W. Kunz n.k. Prhn, \Multi-Lvl Logi Optimiztion Implition Anlsis," Pro. ICCA, pp. 6-13, [9] M. Shulz n E. Auth, \Avn Automti Tst Pttrn Gnrtion n Runn Intition Thniqus," Pro. TCS, pp , [10] E.Sntovih t. l. \SIS: A Sstm or Squntil Ciruit Snthsis" Mmornum UCB M92/41, UC, Brkl.