BDD-BASED LOGIC OPTIMIZATION SYSTEM

Transcription

1 -ASE LOGIC OPTIMIZATION SYSTEM Conun Yn Mij Cisilski urry 2 TR-CSE-- yn,isils.umss.u prtmnt o Eltril n Computr Eninrin Univrsity o Msshustts Amhrst, MA 3

2 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 2 -ASE LOGIC OPTIMIZATION SYSTEM Conun Yn Mij Cisilski yn,isil s.umss.u prtmnt o Eltril & Computr Eninrin Univrsity o Msshustts I. INTROUCTION Loi synthsis plys ntrl rol in th sin utomtion o VLSI iruits. Sotwr tools or loi synthsis r on o th most importnt tools vr vlop in th r o Computr-Ai sin (CA). With th hlp o thos tools, sinr is r rom tious n rror-pron low-lvl iruit sin, n n ous on rhitturl n lorithmi lvl issus. Loi synthsis is ompos o thr min stps. irst, iruit sri in hih-lvl lnu (hrwr sription lnus, suh s VHL or Vrilo) is trnsorm into ooln ntwork. Thn, th ooln ntwork is optimiz usin loi optimiztion tools. inlly, th optimiz ooln ntwork is mpp to lirry o loi lls. Th ntir pross is irt in suh wy s to optimiz rtin sin ojtivs (suh s ly, r, powr, t) n mt usrs spiitions n onstrints. Amon ths thr stps, loi optimiztion is th most importnt. us th qulity o inl synthsis rsults is minly trmin y it. As rsult, intnsiv rsrh hs n on in this r. A. Tritionl Multi-Lvl Loi Optimiztion Th min thm in multi-lvl loi optimiztion is toriztion. In typil loi synthsis nvironmnt, ooln untion is initilly rprsnt s sum-oprout (SOP) or u orm. This orm is trnsorm y torin out ommon lri or ooln xprssions. In n lri toriztion, loi untions r trt s polynomils, in whih ruls o ooln lr r not ppli. ooln toriztions, s on ooln ivision, pply ooln lr ruls, hn n prou ttr rsults in trms o th rsultin loi omplxity (numr o trms, litrls, t). Tritionl loi optimiztion mthooloy, s on lri toriztion or ooln ntworks [], [2], hs in trmnous suss in loi optimiztion n mr s th ominnt mtho. Howvr, whil nr optiml rsults n otin or thos ooln untions whih n rprsnt with AN/OR xprssions, rsults r r rom stistory or untions whih n omptly rprsnt s omintion o AN/OR n XOR xprssions. This work hs n support y rnt rom NS unr ontrt No. MIP Althouh loi optimiztion mthos s on ooln toriztions, n potntilly or ttr rsults thn lri mthos, thy il to ompt with lri mthos u to thir hih omputtionl omplxity. W liv tht th ilur o ooln optimiztion is us y inpproprit t strutur us to rprsnt ooln untions. Cu rprsnttion, whih is riv rom two-lvl AN/OR orm (PLA), nturlly vors lri-s mthos. This rprsnttion, howvr, is not suitl or ooln oprtions. Consquntly, ooln oprtions suh s MUX n XOR riv lss ttntion rom th innin o loi synthsis rsrh.. Nw Opportunity Throuh th ontinuously intnsiv rsrh n vlopmnt in loi synthsis r or th lst twnty yrs, th nrl rmwork or loi synthsis hs n wll stlish. Whil th sp or urthr improvmnt o th synthsis low sms to limit, thr is still potntil or siniint improvmnt in mny prours in synthsis pross [3]. This is spilly tru whn mor iint wys to rprsnt ooln untions om vill. A ri rviw o loi synthsis history is shown in i.. It n rouhly ivi into thr prios, rprsnt y thr most mous mthos: Quin-MClusky n ESPRESSO or two-lvl loi minimiztion, n SIS or multi-lvl loi optimiztion. Quin-MClusky mtho rquirs ooln untion to rprsnt in th mintrm orm. Sin th siz o mintrm rprsnttion is xponntil in th numr o inputs, this mtho is o thortil importn only. ESPRESSO [4], th irst prtil loi minimiztion tool, works on th sum-o-prout (SOP) orm whih is muh mor ompt thn mintrm-s rprsnttion. Th synthsis mtho in this tory ws ltr push to th limit y Court [5] y inorportin impliit numrtion thniqus. inlly, SIS [2] is th most sussul synthsis tool vlop so r. It orms th kon o most morn mi n ommril loi synthsis tools. Th ntrl thm in SIS is lri toriztion in whih tor orm ws us s ooln loi rprsnttion. Compr with th SOP orm, tor orm is muh mor onis n losr to th inl t-lvl implmnttion. SIS still pns on two-lvl orms to rry out loi minimiztion o iniviul nos o ooln ntwork.

3 " ) : : YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 3 Th history o loi synthsis monstrts simpl, yt lr t tht th ooln loi rprsnttion plys ntrl rol in th volution o synthsis mthos. It sms quit nturl tht loi synthsis mthos will kp volvin with th mrn o nwr n mor iint ooln loi rprsnttions. W liv tht th p o this volution is inrsin with th umultion o xprtis in inry ision irms (s). Our rsrh is tryin to rss this nw opportunity. Krnuh mp mintrms Sum-o-prout Cus tor orm s C. Min Contriution Quin-MClusky Esprsso SIS i.. A ri history o loi synthsis. A nw omposition thory is prsnt in this ppr. W show tht loi optimiztion n iintly rri out throuh itrtiv omposition n mnipultion. Our pproh provs to iint or oth AN/OR- n XOR-intnsiv untions. This is th irst unii loi optimiztion mthooloy tht llows to optimiz oth lsss o untions. W lso propos prtil, omplt, -s loi optimiztion systm, S, tht n hnl ritrrily lr iruits. A nrl rmwork whih inorports typil loi synthsis prours hs n implmnt in S. A numr o nw mnipultion thniqus, whih prov vry iint t mnipultin s in th prtition ooln ntwork nvironmnt, r lso prsnt. II. ACKGROUN AN TERMINOLOGY A. ooln untion A ompltly spii ooln untion with -inputs n output is mppin, whr. A ompltly spii ooln untion n uniquly! in y its on-st, in s. Or o-st, in s #"$"% &. or ompltly spii ooln untions n ', is ovr y ', i ( ( *. An inompltly spii ooln untion with -inputs n output is mppin +%,.-, whr Y =,, *, whr * stns or on t r. Th on t r st (-st) o n inompltly spii ooln! ooln untion is in s /( An inompltly spii ooln untion n uniquly in y its st 5( (/, (#"$" /, or 5( (#"$". In th ontxt o this work, w r only onrn with ompltly spii ooln untions. In th squl, ooln untion is rrr to s ompltly spii ooln untion. A. Rprsnttion o ooln untions A ooln untion n rprsnt in mny irnt orms. A orm 7 is si to nonil i th rprsnttion o ooln untion y 7 is uniqu. An xprssion rprsntin ooln untion n riv rom its truth tl y inin th sum o rows (trms) or whih th untion ssums vlu. Th xprssion s on th sum o mintrms is lso rrr to s nonil sum-o-prout orm. Mintrms r ommonly us to rprsnt ooln untions. Howvr, u to th xponntil ntur o this rprsnttions, whih rquirs 89 trms or - input untion, its pplition is limit to simpl ooln untions, n minly us or illustrtion purposs only. A mor prtil rprsnttion o ooln untion is th sum-o-prout (SOP) orm, whih n otin y simpliyin th mintrm-s rprsnttion usin ruls o ooln lr. Eh trm in SOP orm is rrr to s SOP trm (or prout trm). Prtilly th numr o prout trms rquir to rprsnt untion is muh smllr thn th numr o mintrms. Howvr, us th simpliition is not uniqu, th SOP orm is not nonil. In multi-lvl loi synthsis, prout trm is lso ll u, n SOP rprsnttion is rrr to s st o us. ormlly, u is prout o litrls, whr litrl is vril or its omplmnt. Cu rprsnttion orms th kon o ll th loi synthsis systms. Howvr, in th r o omplx, multi-million-t sins, u rprsnttion o ooln untion oms mor n mor imprtil. In th ollowin stions, w shll isuss som othr, mor iint orms to rprsnt ooln untions. A.2 untionl Expnsion n ision irms A nonil rprsnttion o ooln untion n otin throuh vrious untionl xpnsions. inition (Shnnon xpnsion) 2 A ooln untion +:;=<?>?>>?=?>?>>A : < >?>>A >?>?> whr n xprss s C = n E r rrr to s th positiv n ntiv otor o w.r.t. vril. Shnnon xpnsion provis th most unmntl wy to ompos ooln untion. Mny othr, mor G In 854 ool [] irst sri this typ o xpnsion tht ws ltr inorrtly rit to Shnnon. E +!

4 E 8 E : # E 8 # YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 4 nrl omposition mthos n riv rom it. or xmpl, it n vrii sily tht th tritionl mintrm rprsnttion n otin throuh itrtiv Shnnon xpnsion. Shnnon xpnsion lso provis n importnt thortil ountion or inry ision irms (s) [7], [8], whih r th min thm o this ppr. s will introu in Stion II-. inition 2 (Orthonorml Expnsion) A ooln untion n xpn usin n orthonorml sis [9]. Lt,, st o ooln untions suh tht : >, n. Thn ny ooln untion n xpn s,, &: > Th trm is ll th nrliz otor o w.r.t.. Not tht th Shnnon xpnsion is spil s o orthonorml xpnsion whn is sinl vril. It will om lr in th ollowin hptrs tht th omposition s on Shnnon xpnsion is limit to sinl no, whil th omposition s on n orthonorml xpnsion is s on roup o nos. Th lttr on, in most ss, will prou mor iint ompositions. This issu will isuss in til in Stion IV-. A.3 untionl omposition Th purpos o untionl omposition is to rk lr ooln untion into smllr prts, h o whih n implmnt y ooln loi o mnl omplxity. Whil, orin to Kohvi [], untionl omposition is n intrinsi proprty o swithin untions, inin oo omposition is not trivil. untionl omposition hs n on o th most tiv rsrh topis or s. Th prolm o untionl omposition, s in y Ashnhurst [], Roth n Krp [2], n ormult s ollows. inition 3: Th ol o untionl omposition is to in ooln untion,,, suh tht ooln untion 5 n xprss s A?>>?>?! E " " $# " +:;>?>?>? whr &% '% :$>?>>A(%*) # &+ '+ :?>>?>'+, $#.-. I /, th omposition is ll isjuntiv; # othrwis it is onjuntiv. is rrr to s oun st, n is rrr to s r st. Usully th omposition n rmtilly simplii i isjuntiv omposition n oun. Thror, isjuntiv omposition hs n th trt o intnsiv rsrh. Th irst systmti pproh to in isjuntiv omposition ws propos y Ashnhurst []. In his mtho, ll vrils r irst prtition into oun st (Y) n r st (Z). Th ooln untion is thn rprsnt s ooln mtrix (lso ll omposition hrt) y usin th vrils in th oun st n th r st s olumn n row inis rsptivly. A isjuntiv omposition, 5 2,, whr / 4, xists i th numr o istint olumns (ll olumn multipliity) 5 8. A omposition hrt or untion %8+ %8+ '+ is shown in i. 2(), with # '%, n th r st. It th oun st n oun sily tht th numr o istint olumns is 2. As rsult, +A '% n isjuntivly ompos s!,9 % +, whr : +. Howvr, i th vrils %!'+ r prtition suh tht th oun st #, n th r st, th isjuntiv omposition will not oun. Th omposition hrt orrsponin to th lttr s is shown in i. 2(). i. 2. w z x y () ;=<?> 3# y z x w () ;A> Two omposition hrts or untion C<EGIHJLK38M?NHJIKOM QP KNMR P K. Ashnhurst s pproh ws xtn y svrl othr rsrhrs [3], [4]. All ths pprohs r hn lssii s Ashnhurst/Curtis omposition mthos. Th ommon hrtristis o ths mthos is tht thy ll pn on ooln mtrix rprsnttion or on omposition hrts [4]. Th rwk o ths pprohs is ovious. or n -input ooln untion, thr r S 89 mtris or hrts. Thror, ths mthos r not prtil rom th ninrin prsptiv. Roth n Krp [2] propos th irst prtil untionl omposition mtho whih ws ltr rrr to s th Roth-Krp omposition. In thir mtho, ooln untion is rprsnt s st o us. Compr with th rprsnttion s on ooln mtrix, u rprsnttion is muh mor ompt n hs pility to rprsnt lrr ooln untions. Th thniqu us in Roth-Krp omposition is s on th prtitionin o us into omptil lsss. It shoul not tht thr is on to on rltion twn th numr o istint olumns in omposition hrt n omptil lss. Thror, thr is no unmntl irn twn Ashnhurst n Roth-Krp omposition. Howvr, u to mor iint wy to rprsnt ooln untions, Roth-Krp omposition is muh mor iint thn Ashnhurst omposition. In viw o th vlopmnt o untionl omposition, it is intrstin to not tht th iiny o omposition pproh is wll orrlt with th rprsnt-

5 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 5 tion o ooln untions. This simpl osrvtion n xtn to not tht mor iint untionl omposition mtho, my om vill whn ooln rprsnttion orm, mor ompt thn u rprsnttion, n oun. In th ollowin stion, th most iint rprsnttion o ooln untion to t, inry ision irm (), is introu.. inry ision irms Th onpt o inry ision irms (s) ws irst propos y L [7] in 959. L monstrt tht swithin untion n iintly implmnt s sris o lol isions ( ), whr A is tkn i vril is ; n is tkn i is. L point out th vnt o inry ision prorms ovr n lri rprsnttion. H lso point out tht inry ision prorms n us or iruit synthsis 3. In 978, Akrs [8] irst opt th trm inry ision irm. H lso prsnt th irst st o ruls to ru. Howvr, s h not n wily knowl until st o iint oprtors wr propos y rynt [5] in 98. Sin thn, th rsrh n vlopmnt o s hv hiv trmnous vn. Thousns o thnil pprs, rsrh projts, n pks ontriut to th unrstnin n iint mnipultion o s. s hv n ppli to lmost vry spt o VLSI CA. Thy prov to th most iint ooln rprsnttion to t.. Constrution n Rution o In th unmntl work o L n Akrs, no xpliit ssumption hs n m out th vril rorrin. rynt [5] show tht unr ix vril orr, iint lorithms n vis to mnipult s. A unr this rstrition is nrlly rrr to s n orr (O). Th O or ooln untion n onstrut usin itrtiv Shnnon xpnsion. or xmpl, i. 3() shows th O o ooln untion,, with th vril orr,. Eh no o th O orrspons to Shnnon xpnsion w.r.t. sinl vril. Th positiv o-tor omput t ivn no is nrlly rprsnt y - (soli), whil th ntiv o-tor is rprsnt y - (sh). An O is si to ru O (RO) i th ollowin two rution ruls hv n ppli: ) no is rmov i its - n - point to th sm no; Quot rom [7]: It hs n mply lr tht, lthouh ooln rprsnttion o swithin iruits hs n th ountion on whih swithin thory h n uilt, th inhrnt limittions in th ooln lnu sm to iiult hurls to surmount. ooln rprsnttion is lri n hihly systmti, ut so inlxil tht it is powrlss inst ll ut sris-prlll iruits. [...] inry-ision prormmin is our ttmpt o wy to t yon ths limittions. It works wll or omputtion. urthr stuis will rquir to in iint wys o minimizin inry-ision prorms n to mk inry-ision prormmin n instrumnt or iruit synthsis. () i. 3. Rution Ruls. () Th O otin throuh Shnnon xpnsion. () RO. () RO with omplmnt s. 2) surph is rmov i it is isomorphi to nothr surph. Th RO or untion in i. 3() is shown in i. 3(). It shoul not tht ths two rution ruls r impliitly rlt to ooln oprtions. Rul orrspons to ooln simpliition, ; Rul 2 orrspons to simpl toriztion,. Thror, th O rution provis nturl mns or impliit ooln simpliition n toriztion. As rsult, RO is n impliitly tor ooln rprsnttion. rynt [5] prov tht th ooln rprsnttion s on RO is nonil. In th rst o this thsis, RO is rrr to s or short. In ition to th ov rution ruls, th siz o n urthr ru usin onpt o omplmnt s. This onpt ws irst introu y Akrs [8], n ws iintly implmnt y r, Rull n rynt []. silly, omplmnt points to th omplmntry orm o untion ( no)..2 Vril Rorrin It is known tht th siz o is vry snsitiv to th vril orr. A rnom, or rlssly hosn vril orr will rquntly rsult in n xponntil siz o th. A ommon prour to onstrut is s ollows. irst, n initil vril orr is trmin n th is onstrut orin to tht orr. Thn, vril rorrin lorithm is invok to urthr minimiz th siz o s. Svrl huristis hv n propos to provi n initil vril orr. Thy minly pn on th topoloil n vril pnn nlysis in ooln ntwork [7], [8]. Althouh ths huristis hiv siniint improvmnt ovr rnom orrin, th siz o n urthr ru throuh vril rorrin. Mny huristi vril rorrin lorithms hv n propos. Most o ths lorithms pn on unmntl oprtion, jnt vril swppin [9], [2], [2]. Th most iint lorithm, sitin, ws propos y Rull [2]. H lso propos mhnism ll ynmi vril rorrin, whih llows to rorr urin th pross o its onstrution. This () ()

6 > o h o o r r o : YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM pproh prtilly rlivs th so-ll mmory low-up prolm, whih is us y lr intrmit s..3 on t Cr Minimiztion Th prolm rss y on t r minimiztion n stt s ollows. Givn two ompltly spii ooln untions n, with th o-st o in on t r to, in ooln untion, not, in in th rn [ ], suh tht th siz o o is minimum. This prolm hs n prov to NP-omplt [22]. Amon ll th propos huristis to prorm suh oprtion [23], [22], [24], [25], RESTRICT oprtor propos y Court [23], is th most iint on. C. ooln Ntwork A ooln ntwork is irt yli rph (AG); th rprsnttion o its strutur is strihtorwr. Vrious ooln ntwork prsnttions ir minly in th wy thy rprsnt th lol untion prtinin to h ooln no. i. 4() shows th ooln ntwork rprsnttion o SIS [2], in whih th untionlity o h ooln no is rprsnt s st o SOP trms. This rprsnttion is ommonly rrr to s multi-lvl SOP rprsnttion..nms o nms o r o r.nms h nms r () Multi-lvl SOP rprsnttion h h o () Lol rprsnttion i. 4. Th u n rprsnttions o ooln nos in th ooln ntwork. Th untionlity o ooln no n lso rprsnt s, s shown in i 4(). This rprsnttion is known s lol rprsnttion. Compr with th multi-lvl SOP rprsnttion, lol rprsnttion is rltivly r o runny, us th runny inhrnt in th SOP orm hs n rmov urin th pross o onstrution. It lso llows or possil shrin twn irnt ooln nos. Thror rprsnttion my potntilly onsum lss mmory thn SOP. A ooln ntwork n lso rprsnt in lol orm. In lol rprsnttion, ooln ntwork is ollps into st o lol nos, on no pr primry output. Eh lol no pns only on primry inputs. i. 5 shows two irnt lol rprsnttions. In i. 5(), h lol no is rprsnt in two-lvl rom. In i. 5(), h lol no is rprsnt s monolithi. W rr to this rprsnttion r h h s lol rprsnttion. Th vnt o orm oms now ovious. Usully th loi runny m in multi-lvl oniurtion n ompltly rmov y ollpsin th ooln ntwork into two-lvl SOP or lol orms. Howvr, suh rprsnttion is not mnl to lr ooln ntworks, in whih th siz o rprsnttion my low up. This issu will urthr isuss in Chptr VI..nms h h.nms h () Two-lvl SOP rprsnttion () Glol rprsnttion i. 5. Two-lvl u n monolithi rprsnttion o ooln ntwork. III. TERMINOLOGY To ilitt th isussion in th squl, w n to in unmntl trminoloy n vlop si thorms rlt to th irnt oprtions on. inition 4 () A is irt yli rph (AG) rprsntin ooln ( C? untion. It n uniquly in s tupl, = 2, whr is th untion no (root), V is th st o intrnl nos, E is st o s, n, r th trminl nos. inition 5 (L s) Th l is n whih is irtly onnt to trminl no o th. Th st o l s, not, n prtition into, th st o l s onnt to, n, th st o l s onnt to. inition (Pths) is th st o ll pths rom th root to trminl no. is th st o ll pths rom th root to trminl no. is th st o ll pths. An ovious, ut importnt proprty o is tht its st ( ) ins th on-st, (o-st, #"" ) o untion. Spiilly, h pth ( ) rprsnts isjoint u in th on-st (o-st (#"$" ) o. Thorm (Intrnl E Proprty) Evry intrnl lons to t lst on pth n on pth A E. Proo: Th thorm is prov y ontrition. Sin is onnt rph, vry must lon to, i vry ithr or. or n 4 pth pssin throuh lons to, thn ll th nos low n ollps into, so tht A. Hn th ontrition. Sm rsonin pplis to. inition 7 (Cut) A ut (! ) o is prtition o its nos V into isjoint susts n (! ) h h h

7 < ' + YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 7 suh tht root n trminls, ) (V-). A ut nnot ross ny pth mor thn on. A horizontl ut is ut in whih th support 4 o n (V - ) r isjoint. i. shows with svrl possil uts. As sri in th nxt hptr, horizontl uts will usul in prormin th omposition. 3 i.. Vli uts on. IV. THEORY O ECOMPOSITION s hv rwn lot o ttntion rom th loi synthsis n vriition rsrh ommunity. This n ttriut to thir xllnt pility or th onis rprsnttion n iint mnipultion o ooln untions. Howvr, most known omposition mthos mploy s n iint pltorm to rry out tritionl ompositions, suh s Ashnhurst [] n Roth-Krp [2] ompositions, n o not utiliz th ull pility o s. s r unmntlly irnt rom tritionl u orms. In u orm, ooln untion is rprsnt s st o iniviul us. Th rltionship twn irnt us is not lr until rtin ruls o ooln lr r ppli. or xmpl, th t tht xists litrl ommon to two us n is not prnt until som sort o toriztion is ppli. In ontrst to tht, s hv olltiv powr to rprsnt ooln untions, n th rltionship twn irnt pths in (i.. us) is ovious. Thror, inst o prormin tritionl untionl omposition usin s solly s pltorm, omposition mthos spiilly tilor or s shoul vlop. Sin is irt yli rph, in orr to unovr th omposition no in suh olltivly rprsnt ooln untion, som kin o rph trvrsl or struturl nlysis thniqus r nssry. In this hptr, omposition thory whih is s on struturl nlysis is prsnt. A. Prvious Work Th mjority o urrnt omposition mthos rlis on two importnt proprtis o s: ) is us s n iint rprsnttion o ooln support is in s th st o vrils ooln untion pns. 2 4 untion; 2) Th strutur o is impliitly rlt to th omposition hrt us y Ashnhurst omposition []; spiilly, th prtitionin o vrils into oun st n r st is irtly rlt to th vril orrin in th. Th ollowin xmpl illustrts this i. Exmpl : Consir untion %8+ % Th omposition hrt (i. 2()), lin to isjuntiv untionl omposition o this untion, is r-rwn in i. 7(). or th purpos o omprison, th rorr or untion is shown in i. 7(). A ut in th prtitions th vrils into oun st n r st. Noti tht th vril prtitionin is xtly th n sm s %! tht + in i. 7(), with th oun st r st. This mns tht oo vril prtitionin or isjuntiv Ashnhurst omposition n lso otin impliitly throuh vril rorrin. w z x y () w z y x () w oun st ut r st i. 7. omposition hrt n o ooln untion. Consir ut in, whih prtitions st o nos into two sts, n ( ). A no whih is onnt to n in th ut in ll ut-no. A st o ut-nos ssoit with &%! ivn ut is ll ut-no-st. In i. 7(), ut-no-st. An importnt osrvtion is tht th rinlity o th utno-st trmins th totl numr o istint olumns in th omposition hrt. This n xplin s ollows. In i. 7(), ny pth rom th root to trminls must o throuh ithr % or. Thror, i += r trt s olumn inis n %! s row inis, th numr o istint olumns is xtly two. Th omposition pross ins y noin th nos in th ut-no-st. This is shown in i. 8. Th numr o its (vrils) rquir or th noin is ' 5, whr is th rinlity o th ut-no-st. or this xmpl, on it (vril) is suiint. A nw vril, ', is introu. % Th o ' n otin y sustitutin n with thir rsptiv os, s shown in i 8(). This rsults in th inl Ashnhurst omposition, ', whr ' +. Althouh n optiml omposition or th ov untion n oun y th mthos, it is not th s or nrl, omplx ooln untions. u to lk o ritrion or oo ut, ut is usully prorm whn th numr o vrils ov th ut is lss %

8 > ) - ) ) YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 8 w z y x w w y z x w i. 8. Ashnhurst omposition usin. thn som ix vlu, = zw + z w = x + y. Th pplition o ths mthos r hn rstrit to Look-Up-Tl (LUT)- s PGAs, with in th numr o inputs to n PGA lok [2], [27]. W liv tht, with th hlp o struturl nlysis o s, this typ o omposition n xtn to ompositions lin to iint multi-lvl implmnttions. W r lso wr o n pproh in whih is us s n inirt orm to unovr oo ompositions. In [28], sust o sptrl oiints o ooln untion, rprsnt s, is lult. Th is thn ompos throuh th xmintion o rtin proprtis o th sust. Sin th lultion o sptrl oiints is vry xpnsiv, this mtho is potntilly omputtionlly intnsiv. inin n iint multi-lvl rprsnttion o ooln untion y nlyzin th strutur o its ws irst stui y Krplus [29] t th rly s o s. H introu th onpt o - n -omintor 5, whih l to n lri AN/OR omposition. i. 9 illustrts th onpt o - n -omintor. silly, -omintor (-omintor) is no whih lons to vry pth ( ). Th xistn o -omintor (-omintor) llows th to ompos into two prts onjuntivly (isjuntivly). -omintor () -omintor i. 9. Exmpl o - n -omintors introu y Krplus. () - omintor ls to n lri onjuntiv omposition, C < M M. omposition, C< M. () -omintor ls to n lri isjuntiv oth - n -omintors r spil ss o our nrliz omintor, isuss in Stion IV-C () Sin Krplus [29], vry littl work hs n rport in this r. As r s w know, thr hv n t lst two ttmpts to prorm loi optimiztion trtin multi-lvl rprsnttions y nlyzin struturs. rto t l. [3] propos mtho whih prorms hirrhil isjuntiv omposition irtly on. This mtho silly nnotts isjuntiv omposition inhrnt in th strutur. Compr with SIS, thir mtho is str n nrts muh ttr rsults on som iruits. Howvr, thir mtho ils to nrt oo ompositions on s with omplmnt s. Stnion t l. [3] propos nrliz otors ooln ivision n toriztion mtho. Givn ivisor, untion > n writtn s ". Consquntly, ooln ivision is prorm y sttin n >. Th rsult n urthr improv " y rlizin tht, n imply on t r sts to h othr. Howvr, u to lk o iint wy to nrt ooln ivisors, th improvmnt o this mtho ovr SIS is mrinl. Nithr o th ovmntion mthos rss nrl omposition o s onto xprssions involvin XOR loi.. unmntls or ivin into th tils o irnt typs o ompositions, lt us irst provi thortil nlysis o two unmntl ompositions, nmly ooln ivision n ooln sutrtion. All othr typs o ompositions n riv rom ths two. inition 8 (ooln ivision) untion is ooln ivisor o i thr xists untion, ll quotint, suh tht. In [32], ooln ivision is in s, n is ll ooln tor. In our omposition shm, w lwys ssum. To omply with th trms ivision, w ll ooln ivisor, inst o tor. In this ppr, w shll us th trms ooln ivision, onjuntiv ooln omposition, n ooln AN omposition, intrhnly. inition 9 (ooln Sutrtion) untion is ooln sutrtor o i thr xists untion, ll rminr, suh tht. In th squl, w shll us th trms ooln sutrtion, isjuntiv ooln omposition, n ooln OR omposition, intrhnly. Thorm 2: [33] untion is ooln ivisor o i n only i. Proo: I is ooln ivisor o, thn thr xists. On th othr hn, suh tht ;. %. Hr is ny untion ). Exmpl 2: Consir two untions n. Sin th on-st o is ovr y tht o, i.., is ooln ivisor o. untion n ompos s. Th s or untion n r shown in i..

9 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 9 Thorm 3: A untion is ooln sutrtor o i n only i. Proo: Th proo is ul o tht o Thorm 2. inition (Co-torin squn) Consir no in. Th pth rom root to n uniquly in s st o vrils, whr h vril my ppr in tru or omplmnt orm. Suh st o vrils is ll otorin squn. I is trminl no, th list is ll trminl o-torin squn. Lt us now stuy th proprtis o ooln untions n in orr to stisy. Sin is rphil rprsnttion o squn o Shnnon xpnsions o th ooln untion, th pross n rily monstrt y usin squn o o-torin oprtions. n xpn usin Shnnon xpnsion s: () n xpn in th sm wy, w otin (2) Thn i th ollowin two onitions r stisi, y inution, th ov onitions n nrliz to ny o-torin squn. Tht is, i (3) (4) is tru or ny o-torin squn, thn. Whn n r rprsnt s s, to hk whthr is tru, only trminls n to hk to s i onition is stisi. Whn is trminl o-torin squn, or onition to tru, n must stisy th ollowin two onitions, 3 3 whr * stns or on t r. Thorm 4 (ooln ivisor onition) is onjuntiv ooln ivisor o, i or vry trminl o-torin squn or ivn vril orrin, or ny trminl o-. Hn is ooln ivisor o. Thorm 4 provis n iint wy to hk whthr ooln untion is ivisor o nothr ooln untion. As will om lr in th ollowin Proo: Sin torin squn, ( 9 (5) () (7) stions, Thorm 4 provis th thortil ountion or nrliz omintor. In th sm mnnr, th onition or ooln sutrtor n lso ormult. Thorm 5 (ooln sutrtor) is isjuntiv ooln sutrtor o, i vry trminl o-torin squn or ivn vril orrin, Proo: Th proo is ul o tht o Thorm 4. C. AN/OR omposition (8) In this stion, irnt typs o ompositions trtin AN/OR loi omposition r prsnt. C. ooln omposition irst, th most nrl strutur lin to ooln AN/OR omposition is xmin. This strutur is rrr to s nrliz omintor. inition (Gnrliz omintor) Consir ut prtitionin th st o nos o untion into n (V-). Th portion o th in y is opi to orm sprt rph. In tht rph, n is onnt to i in th oriinl o, n it is onnt to i E in th oriinl o. All th intrnl s r lt nlin. Th rsultin rph is ll nrliz omintor. i shows th onstrution o nrliz omintor. In i. (), ut is prorm on th. Thn th portion ov th ut is opi to sprt rph, whih is shown in i. (). Th onstrution is omplt y onntin s o th rph to th orrsponin trminls in th oriinl. Not tht us o th nlin s, nrliz omintor is not. y ssinin th nlin s to irnt onstnt vlu ( or ), n us to ompos onjuntivly or isjuntivly. Lt st o ll nlin s. = + () () i.. Gnrtion o ooln ivisor s on nrliz omintor. Th ollowin thorm shows how to otin ooln ivisor n prorm th ivision y rirtin th nlin s o to onstnt no. Thorm (Constrution o, ) Givn nrliz omintor o untion, th ooln ivisor is otin W lso rr to it s ooln AN omposition

10 : E YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM rom y rirtin nlin s E o to. Th quotint is otin y minimizin with th o-st o s on t r st. Proo: Aorin to Thorm, thr is t lst on pth pssin throuh h intrnl o. y rirtin ths intrnl s o to, th otin (untion) ovrs ll pths, i.. (. Thror, is ooln ivisor or untion (s Thorm 2). Th quotint n vrii y hkin whthr th onition is tru or ll possil pths in th. Rll tht is opy o, xpt or th s whih orrspon to th o-st o. Thror, th on-st o nonminimiz,. Sin (, whnvr. y onstrution,, n is st to on t r, so tht (. Thror, whn. is th quotint o this ooln ivision. Exmpl 3: To illustrt th Thorm, simpl xmpl is shown in i.. Th o untion is shown in i. (). irst, ut is prorm on th. Thn, nrliz omintor is onstrut s on th ut. Sin ooln ivision is ntiipt, th nlin s on th nrliz omintor r rirt to onstnt. Th ooln. Th quotint o ivisor is sily vlut s this ivision n otin rom y sttin th o-st r,. Noti tht " ". o s on t r. Atr minimiztion o with this on t, n = + () C Q () () i.. A simpl xmpl o ooln ivision. Q = + = + In th ollowin, mor omplx xmpl is provi. Exmpl 4: A omplt onjuntiv (AN) omposition, inluin th onstrution o quotint, is shown in i. 2. In i. 2(), ut is prorm in th. In i. 2(), th nrliz-omintor is otin y opyin th portion ov tht ut to rph. Thn ooln ivisor is uilt y rirtin ll th nlin s o tht rph to. Th ru o is lso shown in i. 2(). As init in th iur, this omposition xposs -omintor in, whih ws not prsnt in th oriinl o. Thror, n sily. In i. 2(), quotint ompos s is otin rom y minimizin untion usin untion s on t r. This rsults in. 9'. As rsult o this pross, th whol untion n ompos s 9' 2. ooln sutrtion is th ul s o ooln ivision. Th ollowin is th unmntl thorm or ooln sutrtion. Thorm 7 (Constrution o, ) Givn nrliz omintor o untion, th ooln sutrtor o n otin y rirtin nlin s o to. Th rminr is otin y minimizin usin th on-st o s on t r st. Proo: Aorin to Thorm, thr is t lst on pth pssin throuh h intrnl o. y rirtin ths intrnl s o to, th o th rsultin untion ovrs ll pths, i.. "$" #"$" (( - ( ). Thror, is ooln sutrtor or untion (s Thorm 2). Th rst o th proo is ul o tht o Thorm. urin th pross o inin n optiml ooln AN/OR omposition, ll possil uts shoul xris. Oviously, th numr o possil uts oul vry lr vn or mium siz. Thror, som iltrin mhnism to ru th numr o nit uts shoul vlop. In th ollowin, svrl iltrs hv n intii to isquliy uts whih r invli or runnt. inition 2 (Vli ut) A ut is ll vli i it ontins t lst on. Othrwis, ut is invli. Thorm 8: Only vli uts l to nontrivil ooln omposition. Proo: Consir n invli ut in th. y inition, th nrliz omintor nrt rom th invli ut os not hv ny s. Hn ll trminl s r nlin. Sin ll nlin s r rirt to (), th ooln ivisor (ooln sutrtor) ( ). Ths ss r shown in i. 3() n (). Now onsir vli ut. Sin, som o th trminl s o th nrliz omintor r onnt ( ), whil othrs ( ) r onnt to ( ). Hn th rsultin n ( ) is nontrivil, lin to nontrivil omposition. inition 3 (-Equivlnt Cuts) Two uts r -quivlnt i thy ontin th sm sust o s. inition 4 (-Equivlnt Cuts) Two uts r -quivlnt i thy ontin th sm sust o s. Thorm 9 (istint Cuts) All ooln ivisors otin rom -quivlnt uts r intil. Proo: Consir two uts, n, whih r -quivlnt. In h o th ooln ivisors nrt y thos uts, s r onnt to. All othr s r onnt to. Hn, oth ooln ivisors hv th sm pths rom root to (on-st) n th sm pths rom root to (o-st). Hn, oth ooln ivisors r intil. This is illustrt in i. 3(), (), whih showin tht ut 2 n ut 3 lon to -quivlnt lss, n hn l to intil ooln ivisors. Thorm (istint Cuts) All ooln sutrtors otin rom -quivlnt uts r intil. Proo: Th proo is similr to tht o Thorm 9.

11 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM ru -omintor = + + C Q C minimiz Q = + + Q -omintor () Oriinl untion () Gnrliz omintor n ooln ivisor () Minimizin with o-sts in s on t r i. 2. Otinin tor orm on. p 2 3 x y z 4 () Vrious uts on ru () Trivil ooln ivisor nrt rom ut ru () Trivil ooln sutrtor nrt rom ut ru ru () ooln ivisor nrt rom ut 2 () ooln ivisor nrt rom ut 3 i. 3. Et o ut on th nrtion o ooln ivisor/sutrtor. In onlusion, inin ut n viw s prtitionin o n s, rthr thn prtitionin o nos. Thror, th totl numr o ll possil uts is 8. An in-pth nlysis o strutur rvls tht th tul numr o vli uts is muh smllr. Th numr o vli uts is urthr limit y th ollowin ut proprty. Thorm (Trnsitiv Cut Proprty) Consir no, n its - (or -). A ut ontinin must lso ontins ll othr s spnnin 7 rom pth rom root to. Proo: Th trnsitiv proprty is urnt y th t tht ut nnot ross th sm pth mor thn on. As shown in i. 3(), thr (% + s, n oriint (spn) rom nos whih r on th sm pth to no %. + Thror, ny ut rossin must lso ross n. An is si to spn rom pth i it is inint to no on th pth.

12 > YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 2 Th trnsitiv ut proprty rmtilly rss th numr o possil uts in. Howvr, sin th tul numr o vli uts pns on spii strutur, it is iiult to iv onrt ormul or th totl numr o vli uts. In our pproh w limit our ttntion to horizontl uts. Our xprin shows tht horizontl uts work wll on most s. Unr th worst s, th totl numr o horizontl uts is, whr V is th numr o vrils (lvls o ). In prti, th totl numr o vli horizontl uts is muh smllr thn, us mny uts r ithr -quivlnt or -quivlnt. C.2 Alri omposition Alri omposition is spil s o ooln omposition. u to th importn o th lri omposition n sinss with whih it n intii on, lri AN/OR struturs r rily intii inpnntly o nrliz omintors. Two si struturs lin to n lri AN/OR omposition wr oun y Krplus [29]. Hr w rviw ths struturs n show tht thy r spil ss o our nrliz omintor. inition 5 (-omintor) No whih lons to vry pth is ll -omintor. It shoul not tht th ov inition pplis only to s without omplmnt s ov no. A with -omintor hs n shown in i. 9(). Thorm 2 (Alri AN omposition) Th whih ontins -omintor n lrilly ompos into two onjuntiv prts, i.., ', whr th supports o n ' r isjoint. Proo: i. 4() shows th strutur o -omintor, in whih no lis on ll pth. I ut is prorm irtly ov no, th ooln ivisor nrt rom th nrliz omintor is struturlly intil to th portion o th ov th ut. This is shown in i. 4(). Th quotint o this ivision n otin y rirtin th s to on t r, whih n thn rirt to no. Thn ll nos in prt hv th sm trnsitiv hil,, n th whol prt ollpss into no. This is shown in i. 4(). Sin thr is no ommon support twn n ', th omposition is lri. inition (-omintor) No whih lons to vry pth is ll -omintor. -omintor is ul o -omintor. An xmpl o -omintor is shown in i. 9(). Thorm 3 (Alri OR omposition) Th whih ontins -omintor n lrilly ompos into two isjuntiv prts, i..,. ', whr th supports o n ' r isjoint. Proo: Th proo is similr to tht o Thorm 2. illustrt in i. 5.. XOR omposition It is omposition s on nrliz omintors, sri in th prvious stions, rlis on s. It is intrstin to not rtin proprtis o s. Nmly, s provi n rly vlution o ooln untion. or xmpl, th vlu o untion ( ) n trmin whn ithr or quls to ( ). s o untions tht r minly ompos o AN/OR loi tn to hv mny s. On th othr hn, s o untions popult with XORs hv vry w or no s. Thror, th vlu o untion with XORs is trmin y th rltiv vlus o its vrils. or xmpl, th vlu o untion will only trmin whn vlus o oth vrils n r ivn. It is pprnt tht th omposition whih rlis on s will il on with w s. In this stion, th thniqus trtin XOR-typ omposition o r vlop. In this s th omplmnt s r us to unovr th unrlyin XOR omposition. Th primry ol o introuin omplmnt s ws to ru th mmory us. Intrstinly, w in tht th prsn o omplmnt s in is rlt to XOR omposition. In th squl, w will us XNOR ( ) inst o XOR us XNOR hs mor strihtorwr rprsnttion on s.. Alri XNOR omposition inition 7 (x-omintor) No whih is ontin in vry pth is ll n x-omintor. x-omintor q y r u x i.. Rol o n x-omintor in XNOR omposition. A with n x-omintor is shown in i.. Not tht th inition o x-omintor implis tht thr must xists t lst on omplmnt ov th x-omintor. Othrwis ll th nos ov will ollps into. Thror x-omintors o not xist on s without omplmnt s. Thorm 4 (Alri XNOR omposition) Lt n x-omintor o th o untion. Th o n lrilly ompos s 2, whr is root t, n is th root t th oriinl untion with rpl y onstnt. Proo: i. 7() shows nri with x-omintor. y inition o omplmnt s, th o root t, s shown in i. 7(). Thn th n rprsnt s isjuntion o two prts, s shown in i. 7(). Not tht n r th -omintors in thir rsptiv s. y inin to th o n split into two prts, n y x q r u

13 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 3 Q v C v v v () -omintor strutur () Gnrtion o th ivisor () Gnrtion o quotint Q i. 4. -omintor strutur n its orrsponin omposition. R v v v v C () -omintor strutur () Gnrtion o th sutrtor () Gnrtion o rminr i. 5. -omintor strutur n its orrsponin omposition. v u u u u () () () Whil xhustiv srh or ll possil untions is lrly prohiitiv, st o oo nits or n tt irtly orm strutur, ll nrliz x-omintor, in s ollows. inition 8 (Gnrliz x-omintor) No whih is point to y oth th omplmnt n rulr s is ll nrliz x-omintor. Th omplmnt s ssoit with th nrliz x-omintor r ll XORrlt omplmnt s. i. 7. x-omintor n its omposition in whih is rpl with, untion n ompos s + 2. Exmpl 5: An x-omintor is shown in i.. Aorin to Thorm 4, th n lrilly ompos s %..2 ooln XNOR omposition Th ol o ooln XNOR omposition o untion is to in omposition ' tht will minimiz th ost o its implmnttion. Usully XNOR omposition is prorm on untion in whih oo AN/OR ompositions r unlikly to oun. Thorm 5 (ooln XNOR omposition) or ooln untion, ivn n ritrry ooln untion, thr lwys n xists ooln untion ', suh tht '. +: urthr =< lrilly ompos, rsultin in!: ;. Proo: Th proo is trivil, usin th ollowin ooln trnsormtion. E. MUX omposition ' (9) whr is n ritrry ooln untion, n '. Lt o ontins nrliz x-omintor. y prormin trnsormtion ', th rulr s pointin to r rirt to (us ), n th omplmnt s pointin to r rirt to ( ). In th pross, th trnsormtion rmovs th XOR-rlt omplmnt s pointin to. Th XNOR or o ooln untion n iintly xtrt y rmovin XOR-rlt omplmnt s rom its. Exmpl : iur 8 shows th or iruit rn4-, tst s in th MCNC nhmrk suit. Aorin to inition 8, thr r two nrliz x-omintors in this +:, nmly, n. W illustrt th omposition s on. Its is shown in iur 8(). Th o ' is lso shown in iur 8(). Th o onsists o n x-omintor, n th o ' onsists o - n -omintors. Thror oth o thm is rphil rprsnttion o squn o Shnnon xpnsions. Eh no in n viw s simpl multiplxor (MUX). Tkin MUX

14 ' YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 4 x2 nrliz x-omintors x3 x x x4 x x4 = x2 x3 x x4 xists in vrious ooln untions, thy r spilly ommon in rithmti untions. Thy r rquntly ssoit with XNOR omposition. Exmpl 7: Shown in iur 2 is simpl xmpl o untionl MUX omposition. Nos n ovr ll pths. Susquntly, untion n ompos s ', whr ' srvs s ontrol sinl o th MUX. () () i. 8. XNOR omposition o untion rn4- omposition rrlss o th spii strutur otn ls to poor multi-lvl ooln xprssions. Simpl MUX omposition w.r.t sinl no is only niil whn th ovrlp twn its two o-tors is lss thn rtin thrshol. This s is shown in i. 9. u v v v i. 2. O< () () Exmpl o untionl MUX omposition: C:< M. M, E. untionl MUX i. 9. Simpl MUX omposition Th nrliztion o simpl MUX omposition is rrr to s untionl MUX omposition. In this omposition, th ontrol sinl is untion, inst o sinl vril. untionl MUX omposition otn ls to onis multi-lvl xprssions. Thorm (untionl MUX omposition) Consir strutur, in whih two nos, n, ovr ll pths, whr. Th n thn ompos s ' is otin y rirtin no to, n no to, n n ' r untions ssoit with nos n, rsptivly. Proo: Th proo is similr to tht o Thorm 4. omposition is shown in i. 2. u h v u i. 2. untionl MUX omposition. C < NM. v h Th Similr to th initions o th - n -omintor, this thorm pplis only to s without omplmnt s ov n. Whil th untionl MUX omposition. Linr Expnsion o s In this stion, nrliztion o irnt ompositions sri in prvious stions is stui. It will shown tht ll prvious omposition mthos r spil ss o linr xpnsion to prsnt hr. Our ojtiv is not to ovrrul ll spil-s ompositions; ths ompositions r o prtil importn, us thy r sy to intiy n thir ompositions r strihtorwr. Th purpos o this stion is to in n unrstnin o th unmntls o ooln omposition. i. 22() shows nri. Eh rprsnts n ritrry loi untion, inluin onstnt untions n. Any oul rprsnt in this wy without loss o nrlity. Lt us xmin th omposition o suh strutur :; <>?>>A into st o isjuntiv omponnt s ( ), shown in i. 22(). Eh omponnt onsists o oiint n untion. Not tht th root o h untion plys rol o -omintor in th rsptiv omponnt. Thror, h omponnt n urthr ompos orin to th - omintor strutur. Th inl omposition is shown in i. 22(). : Now < >?>?>? lt us stuy th proprtis o oiint s ( ). Th rltion twn thos oiints r shown in i. 23. Sin ll oiint s r nrt rom th sm, n ir only in thir trminls, ll oiint s r rphilly isomorphi. Aorin to th prinipl o APPLY oprtion [5], th ooln oprtions twn thos oiint s only tk pl t th trminls. Thror, th union o ll oiint s is qul to, whih is shown in

15 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 5 2 k 2 k 2 k () A nri () Linr xpnsion o th 2 k 2 k () omposition o ll omponnts usin -omintor i. 22. Linr xpnsion o i. 23(). Similrly, th intrstion twn ny two irnt oiint s is qul to, whih is shown in i. 23(). Mthmtilly th ov nlysis n ormult s ollows. Thorm 7 (Linr Expnsion) A ooln untion n :$xpn <?>>?> w.r.t. n orthonorml oiint st,, s ollows: L : L : >. whr n Proo: Any ooln untion n rprsnt s with strutur shown in i 22. This iur n i. 23 provi th proo. W not tht our linr xpnsion thory souns xtly lik th inition 2 (orthonorml xpnsion). Howvr, th wy in whih th two xpnsions r rri out is irnt. Whn ooln untion is rprsnt symolilly, in orr to prorm th orthonorml xpnsion, n orthonorml st must provi irst. Gnrtion o suh symoli orthonorml st is not trivil. Also th nrliz o-tors rquir or th orthonorml xpnsion n to lult. Worst o ll, th tivnss o symoli orthonorml xpnsion will not ully roniz until th whol omposition is omplt. In ontrst to tht, th linr xpnsion n prorm sily on, us th oiint s n untion s r rprsnt xpliitly y strutur. Th only thin tht ns to on is to iur out whih st o oiints shoul us or th omposition. Similrly, strutur provis lots o hints or this typ o omposition; som struturl nlysis o is rquir or this purpos. Th tivnss o linr xpnsion n lso rily stimt y th nlysis o th strutur. In summry, Thorm 7 provis urthr lxiility to ompos n ritrry. Th ppliility o this thorm rlis on inin strutur to whih this thorm n ppli iintly. Th spil ss, nmly th -omintor, -omintor, x-omintor, simpl MUX, n untionl MUX omposition, in whih th numr o omponnt s is limit to 2, hv n tkn r o in th prvious stions. Th struturs mor nrl thn prviously in omintors shoul intii. W ntiipt tht this nrliztion will urthr improv th prormn o our omposition shm. V. LOGIC SYNTHESIS ASE ON ECOMPOSITION - LOPT In this stion, implmnttion tils o th loi optimiztion prorm, lopt, whih is s on our omposition thory, r prsnt. Alorithmi nlysis o prours in th propos loi synthsis low is lso provi. It will shown tht ll nssry prours in typil loi optimiztion low n implmnt throuh sris o mnipultions n ompositions. or xmpl, ooln simpliition n iintly rri out throuh vril rorrin; toriztion n on throuh rursiv ompositions; n loi shrin n iintly tt on th inl torin trs. A. Synthsis low Th synthsis low or lopt is outlin in i. 24. Th low onsists o two mjor prts, omposition n torin tr prossin. irst, th lol s (s Stion II-C) r onstrut or th ooln ntwork. Thn, th lol s r sumitt to th omposition nin or loi omposition. Alon with th omposition, st o torin trs r onstrut to ror th omposition. In th pross o omposition, lr is rursivly ompos into sml prts. Th omposition pross stops whn

16 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 2 k () Sum o ll oiints quls. i j i j () Intrstion twn ny two irnt oiints quls. i. 23. Coiint proprtis ooln Ntwork Construt Glol s omposition Enin Construt On No on th torin Tr N hs on no? Y torin Tr Prossin Synthsis Rsult Prsnttion Thnoloy Mppin i. 24. Synthsis low o lopt hs on no. inlly, n importnt prour, shrin xtrtion, tks pl in th torin tr prossin phs. us o its itrtiv ntur, th ovrll omplxity is iiult to hrtriz. In th xprimnt, w will ous on th run tim omprisons with th stt-o-th-rt loi synthsis prorm, SIS.. stor/lo Mhnism In this stion, mnipultion thniqu, whih is ruil to prormin loi simpliition in lopt, is xplin. In our -s loi optimiztion shm, vril rorrin lorithm srvs s n impliit loi simpliition. It shoul mphsiz tht, in typil pk, vrils r rorr with rspt to mnr, n not w.r.t. spii. Hn, i thr is mor thn on in th mnr, vril rorrin my not rsult in th sir simpliition or spii. In orr to hiv mximum loi simpliition o ooln untion (), ll othr s must r rom this mnr or prormin vril rorrin. Howvr, thos r s must prsnt in th mnr whn thy r n or omposition t ltr tim. Thror, n iint stor/lo mhnism must vlop. A niv wy o storin is to ump it into SOP orm. Th vnt o SOP orm is tht th n ronstrut unr vril orr whih is irnt rom th orr in whih th is stor. This ors som lxiility or th implmnttion. Howvr, sin th numr o SOP trms o n xponntil in th numr o nos, storin s in SOP orm is not sil solution. A nw t strutur, Pool, hs n vis to prorm stor/lo oprtions. silly, Pool is AG whih is rphilly isomorphi to th it rprsnts. A is opi to Pool or it is r rom th mnr. Th n ronstrut ltr y pplyin n it 8 oprtion tims, whr is th numr o nos. Sin n it oprtion tks onstnt tim, th ovrll omplxity o our stor/lo lorithm is S. Th isvnt o Pool is tht th vril orr o th mnr into whih is lo must th sm s th orr in whih is stor. orin mnr to rtin vril orr oul rsult in n xponntil inrs in th siz i th mnr is not mpty. An it is short or i-thn-ls; it is in s H < H M H

17 ' < < YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 7 Howvr, in our pplition, whn is lo (uilt), th mnr is lwys mpty. Anothr importnt tur o our Pool mhnism is to llow th vril sustitution urin th pross o ronstrution. This n omplish sily y moiyin th it oprtor s, whil is th mppin o vrils. This tur plys ruil rol in our iint itrtiv limint prim (Stion VI- C). C. Th omposition Enin Shown in Alorithm is th min prour or th omposition. To mk th mnr ln, ll s r stor in th Pool orm. A is lo into th mnr or it is ompos. Th stor/lo pross is rliz y untion stor n lo. Atr hs n onstrut in th mnr, it is ompos y ompos. Th omposition rsults r prsnt s ', whr stns or ooln oprtor, suh s AN, OR, XOR or XNOR. Th omposition is stor in th orm o torin tr, isuss in Stion V-. Th intrmit s o ' n r thn stor in th Pool orm n nquu i thy hv mor thn on no. Th omposition pross is itrt until th quu is mpty. Pool = stor(); Enquu(Q, Pool); whil(pool = quu(q)) { = lo(pool); (, h, op) = ompos(); onstrut on no on torin tr; i (!= sinl no) { Pool = stor(); Enquu(Q, Pool) } i (h!= sinl no) { hpool = stor(h); Enquu(Q, hpool); } } rturn (torin tr); Alorithm : Min omposition low Th min omposition nin, ompos, is srh pross or th most iint omposition, rom mor iint (lri) to lss iint (ooln). Th omintors r mpirilly orr in trms o omposition iiny s ollows: ) simpl omintor (-, - n x-omintor), 2) untionl MUX, 3) sinl MUX, 4) nrliz omintor n 5) nrliz x- omintor. inlly, i ll srhs il, th is ompos usin otor w.r.t. th top vril. In prti, th lst stp is rrly rh. It is put hr to nsur th will ompos whn ll othr ttmpts il. A omposition pross ins with th struturl sn, in whih th struturl inormtion o is otin. Th inormtion is us s uin or ll th ollowin ompositions. In trms o ritility o omputtionl omplxity, untion Sn is th most importnt on, us it is ll vry tim th omposition nin is invok. Th thniqu vlop or this purpos is s on mrkin. Th omplxity o Sn is S 5, whr is th numr o nos, is th numr o vrils in th. C. Simpl omintor All thr simpl omintors (-, - n x-omintor) shr similr pttrn, i.., thr is no into whih ll intrnl s onvr. s on this osrvtion, iint lorithms n sin to unvil ll simpl omintor struturs. In t, th sn prour Sn is vis to in out th struturl inormtion o. Th struturs o simpl omintors r lry no in th t ollt y Sn. Th omplxity o this untion is S. In implmnttion, inst o rturnin th irst oun simpl omintor, ll simpl omintors r otin n th on losst to th mil hiht o th is rturn. This hlps to hiv mor ln omposition, whih is ruil to th ly minimiztion. C.2 Gnrliz omintors n Gnrliz x-omintors I n lri omposition os not xist or ooln untion, ooln omposition will prorm. Th struturs lin to ooln ompositions r nrliz omintors n nrliz x-omintors. Unlik th ompositions s on simpl omintors, whos omposition rsults r wll-in, th ompositions s on nrliz omintors rly on minimiztion w.r.t. on t r. Thror, th omposition rsult pns on th iiny o th on t r minimiztion lorithms. To rry out ths ompositions, ll possil ooln ompositions r xmin lvl y lvl. sn inormtion is rquir or th pplition o vrious iltrs. On h lvl, two mjor stps, nrliz omintor nrtion n minimiztion w.r.t. on t r, r involv in sinl omposition. Th nrtion o nrliz omintor is pross o opyin th strutur ov th ut. Th uppr oun or this oprtion is S. Th untion us to lult (or ) is s on RESTRICT oprtor [34] whos omplxity is > S > (or S ). Th uppr oun or RESTRICT is S 5. Thror, uppr oun or untion omp- Gnrlizomintor is. Constrution n Prossin o torin Trs A torin tr is wy to ror omposition pross. or xmpl, i ooln untion is.

18 YANG AN CIESIELSKI: -ASE LOGIC OPTIMIZATION SYSTEM 8 ompos into ', thn nw no, with oprtor + n two silins, ' n, will rt to ror this omposition. A torin tr will kp rowin until th omposition is omplt. Susquntly, svrl stps n ppli to th torin trs to urthr optimiz th synthsis rsults. In prtiulr, shrin twn irnt torin trs n iintly tt. To intiy th shrin twn irnt torin trs, s r onstrut or ll torin trs in ottomup shion. Th noniity proprty o is us to intiy untionlly quivlnt su-trs. i. 25 shows n xmpl o shrin xtrtion on tst s.li rom MCNC nhmrk st. E. Exprimntl Rsults Th xprimnts wr onut on SUN UltrSPARC- 5/32M. Thy ovr most o th omintionl tst ss rom th MCNC nhmrk st. All th tst ss n rouhly toriz into two roups: ) AN/OR-intnsiv untions, n 2) XOR-intnsiv loi (rithmti untions). Th litrl ount or ompositions nrt y lopt ws ompr with th numr o litrls in th tor orm otin y SIS-.2 runnin sript.ru. Th omprison lso inlus rsults tr thnoloy mppin. oth trs SIS mppr n ooln mthin-s rs [35] r us. rs is s on ooln mthin rthr thn tr mthin. or this rson th XOR ompositions oun y lopt r likly to prsrv. Th rsults or AN/OR intnsiv iruits r shown in Tl I. On vr, lopt uss slihtly wr ts thn SIS, n mor r thn SIS. Th sliht inrs in r is u to th hihr ost o XOR ts implmnt in CMOS. On vr, th inl synthsis rsults usin lopt n SIS on this lss o untions r lmost th sm. Whil nr optiml rsults r otin y oth SIS n lopt, ut lopt out-prorms SIS rmtilly in CPU tim. Howvr, or th lss o rithmti untions n XOR-intnsiv loi, shown in Tl II, lopt outprorms SIS in ll spts. Whil, in prinipl, rs nrts ttr mppin rsults tht SIS mppr, it ws not stl on svrl iruits whih mks th omplt omprison iiult. or this rson only rsults o SIS mppr r prsnt. Th rsults o thniqus trtin spiilly XOR omposition y Tsi t l [3] r lso list or omprison purpos. On n s tht th prormn o lopt in trms o th numr o ts is omprl to tht o Tsi t l. [3]. It shoul not tht mny XORs in th ntlist synthsiz y lopt r lost tr thnoloy mppin. As init in olumn XORs in Tl II, only XORs r prsrv in thnoloy mppin. VI. -ASE LOGIC SYNTHESIS SYSTEM - S A vry importnt tur o loi synthsis systm is its slility. Th slility rquirs tht th siz o th rprsnttion o prolm proportionl to th siz o th prolm itsl. In our s, th siz o shoul proportionl to th siz o iruit (whih is ommonly msur y th numr o loi ts). Howvr, th siz o lol s or ivn ooln ntwork is ompltly unpritl. It stronly pns on th typ o th iruit, rthr thn on th totl numr o ts. Rprsntin th ntir ooln ntwork y lol s uss srious omputtionl prolms. Thror, propr prtitionin o th ooln ntwork is rquir prior to prormin th omposition. Tl III shows th omprison o th siz o lol s n lol s (in in Stion II-C). It n oun tht th siz o lol s oul s muh s two orrs o mnitu lrr thn lol s. Th similr prolm, lr two-lvl rprsnttion, hs lso n osrv in tritionl multi-lvl loi synthsis. ortuntly, propr wy to hnl it hs n oun. Givn lr ooln ntwork, its multilvl strutur shoul prsrv s muh s possil. Th numr o SOP trms oul too lr or th loi optimiztion lorithms i th ntir ooln ntwork is ollps into two-lvl orms. rom this point o viw, th ntwork prtitionin y -s loi synthsis is similr to th on y tritionl multilvl loi synthsis. Ciruits Glol s Lol s C C C C C C C C C88 54 pir rot TALE III COMPARISON O NUMER O NOES OR GLOAL AN LOCAL CONSTRUCTION In this stion, nw loi synthsis systm, S, whih hs th pility to optimiz ritrrily lr iruits, is prsnt. A. Synthsis low Currnt multi-lvl loi synthsis low xmplii y SIS hs rwn rom ovr twnty yrs o intnsiv rsrh. W liv it hs th pility to hnl vry lr iruits n it os rsp th ssn o loi synthsis in nrl. Thror, S opts th nrl synthsis low o SIS. i. 2 omprs th synthsis low o SIS n S. Th similrity twn thm is ovious. Th unmntl irn twn SIS n S is th wy in whih h systm rprsnts ooln