BDD-BASED LOGIC OPTIMIZATION SYSTEM

Transcription

1 BDD-BASED LOGIC OPTIMIZATION SYSTEM Conun Yn Mij Cisilski urry 2 TR-CSE-- yn,isil@s.umss.u Dprtmnt o Eltril n Computr Eninrin Univrsity o Msshustts Amhrst, MA 3

2 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 2 BDD-BASED LOGIC OPTIMIZATION SYSTEM Conun Yn Mij Cisilski Dprtmnt o Eltril& Computr Eninrin Univrsity o Msshustts I. INTRODUCTION 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 Dsin(CAD). Withthhlpothostools, sinrisrrom 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 VHDL or Vrilo) is trnsorm into Booln ntwork. Thn, th Booln ntwork is optimiz usin loi optimiztion tools. inlly, th optimiz Booln 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. Bus th qulity o inl synthsis rsults is minly trmin y it. As rsult, intnsiv rsrhhsnoninthisr. A. Tritionl Multi-Lvl Loi Optimiztion Th min thm in multi-lvl loi optimiztion is toriztion. In typil loi synthsis nvironmnt, Booln untion is initilly rprsnt s sum-oprout(sop) or u orm. This orm is trnsorm y torin out ommon lri or Booln xprssions. In n lri toriztion, loi untions r trt s polynomils, in whih ruls o Booln lr r not ppli. Booln toriztions, s on Booln ivision, pply Booln 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 Booln ntworks[],[2], hs in trmnous suss in loi optimiztion n mr s th ominnt mtho. Howvr, whil nr optiml rsults n otin or thos Booln untions whih n rprsnt with AND/OR xprssions, rsults r r rom stistory or untions whih n omptly rprsnt s omintion o AND/OR n XOR xprssions. ThisworkhsnsupportyrntromNSunrontrtNo. MIP Althouh loi optimiztion mthos s on Booln 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 Booln optimiztion is us y inpproprit t strutur us to rprsnt Booln untions. Cu rprsnttion, whih is riv rom two-lvl AND/OR orm(pla), nturlly vors lri-s mthos. This rprsnttion, howvr, is not suitl or Booln oprtions. Consquntly, Booln oprtions suh s MUX n XOR riv lss ttntion rom th innin o loi synthsis rsrh. B. 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 thsynthsislowsmstolimit,thrisstillpotntil or siniint improvmnt in mny prours in synthsis pross[3]. This is spilly tru whn mor iint wys to rprsnt Booln 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 Booln 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 tororm wsussboolnloirprsnttion. ComprwiththSOPorm,torormismuhmor onis n losr to th inl t-lvl implmnttion. ½ SISstillpnsontwo-lvlormstorryoutloiminimiztion o iniviul nos o Booln ntwork.

3 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 3 Th history o loi synthsis monstrts simpl, yt lr t tht th Booln 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 Booln loi rprsnttions. W liv tht th p o this volution is inrsin with th umultion o xprtis in Binry Dision Dirms(BDDs). Our rsrh is tryin to rss this nw opportunity. Krnuh mp mintrms Sum-o-prout Cus tor orm BDDs C. Min Contriution Quin-MClusky Esprsso SIS i.. A ri history o loi synthsis. A nw BDD omposition thory is prsnt in this ppr. W show tht loi optimiztion n iintly rri out throuh itrtiv BDD omposition n mnipultion. Our pproh provs to iint or oth AND/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, BDD-s loi optimiztion systm, BDS, tht n hnl ritrrily lr iruits. A nrl rmwork whih inorports typil loi synthsis prours hs n implmnt in BDS. A numr o nw BDD mnipultion thniqus, whih prov vry iint t mnipultin BDDs in th prtition Booln ntwork nvironmnt, r lso prsnt. II. BACKGROUND AND TERMINOLOGY A. Booln untion A ompltly spii Booln untion with Ò-inputs n ½outputismppin Ò,whr ¼ ½. AompltlyspiiBoolnuntionn uniqulyinyitson-st ÓÒ,ins ÓÒ Ü Üµ ½.Oro-st,ins Ó Ü Üµ ¼. orompltlyspiiboolnuntions n, is ovry,i ÓÒ ÓÒ. An inompltly spii Booln untion with Ò-inputs n ½outputismppin Ò,whrY=,, *,whr*stnsoron tr.thon trst(-st)o ninompltlyspiiboolnboolnuntion ܵ isins Ü Üµ. Aninompltly spii Booln untion n uniquly in y its st ÓÒ µ, Ó µ,or ÓÒ Ó µ. Inthontxtothiswork,wronlyonrnwith ompltly spii Booln untions. In th squl, Booln untion is rrr to s ompltly spii Booln untion. A. Rprsnttion o Booln untions A Booln untion n rprsnt in mny irntorms. Aorm issitononilith rprsnttionoboolnuntiony isuniqu. An xprssion rprsntin Booln untion n rivromitstruthtlyininthsumorows (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 Booln untions. Howvr, u to th xponntil ntur o this rprsnttions, whih rquirs Ò trms or Ò- input untion, its pplition is limit to simpl Booln untions, n minly us or illustrtion purposs only. A mor prtil rprsnttion o Booln untion is th sum-o-prout(sop) orm, whih n otin y simpliyin th mintrm-s rprsnttion usin rulsoboolnlr. EhtrminSOPormis rrrtossoptrm(orprouttrm).prtillyth 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 llu,nsoprprsnttionisrrrtos stous.ormlly,uisproutolitrls,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 Booln untion oms mor n mor imprtil. In th ollowin stions, w shll isuss som othr, mor iint orms to rprsnt Booln untions. A.2 untionl Expnsion n Dision Dirms A nonil rprsnttion o Booln untion n otin throuh vrious untionl xpnsions. Dinition(Shnnonxpnsion) 2 ABoolnuntion Ü ½ Ü Ü Ü Ò µnxprsss Ü ½ Ü Ü Ü Ò µ Ü ½ ܼ ¼ whr ½ n ¼ rrrrtosthpositivnntiv otoro w.r.t.vril Ü. Shnnon xpnsion provis th most unmntl wy to ompos Booln untion. Mny othr, mor In854Bool[6]irstsrithistypoxpnsionthtwsltr inorrtly rit to Shnnon.

4 YANG AND CIESIELSKI: BDD-BASED 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 Binry Dision Dirms(BDDs)[7],[8],whihrthminthmothis ppr. BDD s will introu in Stion II-B. Dinition 2(Orthonorml Expnsion) A Booln untion n xpn usin n orthonorml sis[9]. Lt, ½,stoBoolnuntionssuhtht È ½ ½,n ½.Thn nyboolnuntion nxpns, ½ Thtrm isllthnrlizotoro w.r.t.. Not tht th Shnnon xpnsion is spil s oorthonormlxpnsion whn is sinl vril. It will om lr in th ollowin hptrs tht th omposition s on Shnnon xpnsion is limit to sinlbddno,whilthompositionsonn orthonorml xpnsion is s on roup o BDD nos. Th lttr on, in most ss, will prou mor iint ompositions. This issu will isuss in til in Stion IV-. A.3 untionl Domposition Th purpos o untionl omposition is to rk lr Booln untion into smllr prts, h o whih n implmnt y Booln 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. Dinition 3: Th ol o untionl omposition is to in Boolnuntion, µ,suhthtboolnuntion µ n xprss s µ ¼ µ ½ µ µ µ whr Ü ¼ Ü ½ Ü Ò Ý ¼ Ý ½ Ý Þ ¼ Þ ½ Þ Ñ.I,thomposition isllisjuntiv;othrwisitisonjuntiv. isrrrto sounst,n isrrrtosrst. 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)nrst(z).thboolnuntionisthn rprsnt s Booln mtrix(lso ll omposition hrt)yusinthvrilsinthounstnthr st s olumn n row inis rsptivly. A isjuntiv omposition, µ µ µ,whr, xists i th numr o istint olumns(ll olumn multipliity). Aompositionhrtoruntion Û ¼ Ü ¼ Þ ¼ ÛÜ ¼ Þ Û ¼ ÝÞ ÛÝÞ ¼ isshownini.2(),with thounst Û Þ,nthrst Ü Ý.It n oun sily tht th numr o istint olumns is2. Asrsult, nisjuntivlyomposs Û Þµ Ü Ýµ Ü ¼ ¼Ý,whr ÛÞ Û ¼ Þ ¼. Howvr, i th vrils r prtition suh tht th oun st Ý Þ, nthrst Ü Û, th isjuntiv omposition will not oun. Th omposition hrt orrsponin to th lttr s is shown in i. 2(). w z x y () y z x w () i.2. Twoompositionhrtsoruntion Û ¼ Ü ¼ Þ ¼ ÛÜ ¼ Þ Û ¼ ÝÞ ÛÝÞ ¼. 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 Booln mtrix rprsnttion or on omposition hrts[4]. Th rwk o ths pprohs is ovious.orn Ò-inputBoolnuntion,thrr Ç Ò µ 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, Booln untion is rprsnt s st o us. Compr with th rprsnttion s on Booln mtrix, urprsnttionismuhmoromptnhspility to rprsnt lrr Booln untions. Th thniqu us in Roth-Krp omposition is s on th prtitionin o us into omptil lsss. It shoul notthtthrisontoonrltiontwnthnumroistintolumnsinompositionhrtn omptil lss. Thror, thr is no unmntl irn twn Ashnhurst n Roth-Krp omposition. Howvr, u to mor iint wy to rprsnt Booln untions, Roth-Krp omposition is muh mor iint thn Ashnhurst omposition. In viw o th vlopmnt o untionl omposition,itisintrstintonotthtthiinyoomposition pproh is wll orrlt with th rprsnt-

5 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 5 tion o Booln untions. This simpl osrvtion n xtn to not tht mor iint untionl omposition mtho, my om vill whn Booln rprsnttion orm, mor ompt thn u rprsnttion, n oun. In th ollowin stion, th most iint rprsnttionoboolnuntiontot,binrydisiondirm(bdd), is introu. B. Binry Dision Dirms Th onpt o inry ision irms(bdds) ws irst propos y L[7] in 959. L monstrt tht swithin untion n iintly implmnt s srisololisions(ì Ü ),whrais tkn ivril Üis;nBis tkn i Üis. Lpoint out th vnt o inry ision prorms ovr n lri rprsnttion. H lso point out tht inry isionprormsnusoriruitsynthsis 3. In 978, Akrs [8] irst opt th trm inry ision irm. H lso prsnt th irst st o rulstorubdd.howvr, BDDshnotn wily knowl until st o iint oprtors wr propos y Brynt [5] in 986. Sin thn, th rsrh n vlopmnt o BDDs hv hiv trmnous vn. Thousns o thnil pprs, rsrh projts, n BDD pks ontriut to th unrstnin n iint mnipultion o BDDs. BDDs hv n ppli to lmost vry spt o VLSI CAD. Thy prov to th most iint Booln rprsnttion to t. B. Constrution n Rution o BDD InthunmntlworkoLnAkrs,noxpliit ssumption hs n m out th vril rorrin. Brynt [5] show tht unr ix vril orr, iint lorithms n vis to mnipult BDDs. ABDDunrthisrstritionisnrllyrrrtosn orr BDD(OBDD). Th OBDD or Booln untion n onstrut usin itrtiv Shnnon xpnsion. or xmpl, i. 3() shows th OBDD o Booln untion, ¼ ¼ ¼,withthvrilorr,.Eh no o th OBDD 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). AnOBDDissitoruOBDD(ROBDD)ith ollowin two rution ruls hv n ppli: ) no Úisrmoviits-n-pointtothsmno; Quotrom[7]: Ithsnmplylrtht,lthouhBooln rprsnttion o swithin iruits hs n th ountion on whih swithin thory h n uilt, th inhrnt limittions in th Booln lnu sm to iiult hurls to surmount. Booln rprsnttion is lri n hihly systmti, ut so inlxil tht it is powrlss inst ll ut sris-prlll iruits. [...] Binry-ision prormminisourttmptowytotyonthslimittions.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. BDD Rution Ruls.() Th OBDD otin throuh Shnnon xpnsion.() ROBDD.() ROBDD with omplmnt s. 2)surphisrmoviitisisomorphitonothr surph.throbddoruntion ini.3()isshown ini.3(). Itshoulnotthtthstworution ruls r impliitly rlt to Booln oprtions. Rul orrsponstoboolnsimpliition, ¼ ; Rul2orrsponstosimpltoriztion, µ. Thror, th OBDD rution provis nturl mns or impliit Booln simpliition n toriztion. As rsult, ROBDD is n impliitly tor Booln rprsnttion. Brynt[5] prov tht th Booln rprsnttion s on ROBDD is nonil. Inthrstothisthsis,ROBDDisrrrtosBDD or short. Initiontothovrutionruls,thsizo BDD n urthr ru usin onpt o omplmnt s. This onpt ws irst introu y Akrs[8], n ws iintly implmnt y Br, Rull n Brynt[6]. Bsilly, omplmnt points to th omplmntry orm o untion(bdd no). B.2 Vril Rorrin ItisknownthtthsizoBDDisvrysnsitiv to th vril orr. A rnom, or rlssly hosn vril orr will rquntly rsult in n xponntil sizothbdd.aommon prourto onstrut BDD is s ollows. irst, n initil vril orr is trmin n th BDD is onstrut orin to tht orr. Thn, vril rorrin lorithm is invok to urthr minimiz th siz o BDDs. Svrl huristis hv n propos to provi n initil vril orr. Thy minly pn on th topoloil n vril pnn nlysis in Booln ntwork[7],[8]. Althouh ths huristis hiv siniint improvmnt ovr rnom orrin, th sizobddnurthrruthrouhvril 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 BDD to rorr urin th pross o its onstrution. This () ()

6 o h o o r r o YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 6 pproh prtilly rlivs th so-ll mmory low-up prolm, whih is us y lr intrmit BDDs. B.3 Don t Cr Minimiztion Th prolm rss y BDD on t r minimiztion n stt s ollows. Givn two ompltly spii Boolnuntions n,withtho-sto inon t rto,inboolnuntion,not,inin thrn[ ¼ ],suhthtthsizobddo 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. Booln Ntwork A Booln ntwork is irt yli rph(dag); th rprsnttion o its strutur is strihtorwr. Vrious Booln ntwork prsnttions ir minly in th wy thy rprsnt th lol untion prtinin to h Booln no. i. 4() shows th Booln ntwork rprsnttion osis[2],inwhihthuntionlityohboolnno isrprsntsstosoptrms.thisrprsnttionis 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 BDD rprsnttion i.4. ThunBDDrprsnttionsoBoolnnosinth Booln ntwork. Th untionlity o Booln no n lso rprsnt s BDD, s shown in i 4(). This rprsnttion is known s lol BDD rprsnttion. Compr with th multi-lvl SOP rprsnttion, lol BDD rprsnttion is rltivly r o runny, us th runny inhrnt in th SOP orm hs n rmov urin th pross o BDD onstrution. It lso llows or possil shrin twn irnt Booln nos. Thror BDD rprsnttion my potntilly onsum lss mmory thn SOP. ABoolnntworknlsorprsntinlol orm. In lol rprsnttion, Booln ntwork is ollpsintostololnos,onnoprprimry 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 BDD. W rr to this rprsnttion r h h s lol BDD rprsnttion. Th vnt o BDD orm oms now ovious. Usully th loi runny m in multi-lvl oniurtion n ompltly rmov y ollpsin th Booln ntwork into two-lvl SOP or lol BDD orms. Howvr, suh rprsnttion is not mnl to lr Booln 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 BDD rprsnttion i. 5. Two-lvl u n monolithi BDD rprsnttion o Booln 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 BDD. Dinition4(BDD)ABDD is irt yli rph (DAG) rprsntin Booln untion. It n uniquly instupl,bdd= Î ¼ ½ µ,whr isth untionno(root),visthstointrnlnos,eissto s,n,rthtrminlnos. Dinition5(Ls)Thlisn whihisirtlyonnttotrminlnoothbdd.th stols,not,nprtitioninto ¼,thsto lsonntto,n ½,thstolsonnt to. Dinition6(Pths) ¼ isthstollpthsromthroot totrminlno. ½ isthstollpthsromthrootto Ë trminlno. ¼ ½ isthstollpths. Anovious,utimportntproprtyoBDDistht itsst ½ ( ¼ )insthon-st, ÓÒ (o-st, Ó )o untion. Spiilly,hpth Ô ½ (Ô ¼ ) rprsntsisjointuinthon-st ÓÒ (o-st Ó ) o. Thorm (Intrnl E Proprty) Evry intrnl µlonstotlstonpth Ô ½ ½ nonpth Ô ¼ ¼. Proo: Th thorm is prov y ontrition. Sin BDDisonntrph,vrymustlonto ithr ¼ or ½. orn µ, ivry pth Ôpssinthrouh lonsto ½,thnllthnos low nollpsinto,sotht ½.Hnth ontrition.smrsoninpplisto ¼. Dinition7(Cut)A ut ( Î ) o BDD is prtitionoitsnosvintoisjointsustsdn(î ) h h h

7 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 7 suhthtroot Dntrminls, (V-D).Autnnot rossnypth Ô morthnon.ahorizontlutisut inwhihthsupport 4 odn(v-d)risjoint. i.6shows BDDwithsvrlpossiluts. As sri in th nxt hptr, horizontl uts will usul in prormin th BDD omposition. 3 i.6. VliutsonBDD. IV. THEORY O BDD DECOMPOSITION BDDs 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 Booln untions. Howvr, most known BDD omposition mthos mploy BDD 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 BDDs. BDDs r unmntlly irnt rom tritionl u orms. In u orm, Booln untion is rprsnt s st o iniviul us. Th rltionship twn irnt us is not lr until rtin ruls o Booln lr r ppli. or xmpl, th t tht xists litrl ommon totwous n isnot prnt until som sort o toriztion is ppli. In ontrst to tht, BDDs hv olltiv powr to rprsnt Booln untions, n th rltionship twn irnt pths in BDD(i.. us) is ovious. Thror, inst o prormin tritionl untionl omposition usin BDDs solly s pltorm, omposition mthos spiilly tilor or BDDs shoul vlop. Sin BDDisirtylirph, inorrtounovrth omposition no in suh olltivly rprsnt Booln untion, som kin o rph trvrsl or struturl nlysis thniqus r nssry. In this hptr, BDD omposition thory whih is s on BDD struturl nlysis is prsnt. A. Prvious Work Th mjority o urrnt BDD omposition mthos rlis on two importnt proprtis o BDDs: ) BDD is us s n iint rprsnttion o Booln supportisinsthstovrilsboolnuntionpns. 2 4 untion; 2) Th strutur o BDD is impliitly rlt to th omposition hrt us y Ashnhurst omposition []; spiilly, th prtitionin o vrilsintoounstnrstisirtlyrltto th vril orrin in th BDD. Th ollowin xmpl illustrts this i. Exmpl:Consir untion Û ¼ Ü ¼ Þ ¼ ÛÜ ¼ Þ Û ¼ ÝÞ ÛÝÞ ¼. Thompositionhrt(i.2()),lin to isjuntiv untionl omposition o this untion, is r-rwn in i. 7(). or th purpos o omprison, th rorrbddoruntion isshownini.7(). Aut inthbddprtitionsthvrilsintoounstn r st. Noti tht th vril prtitionin is xtly th sm s tht in i. 7(), with th oun st Þ Û n rst Ý Ü. Thismnsthtoovrilprtitionin or isjuntiv Ashnhurst omposition n lso otin impliitly throuh BDD vril rorrin. w z x y () w z y x () w Boun st ut r st i. 7. Domposition hrt n BDD o Booln untion. ConsirutinBDD,whihprtitionsstoBDD nos Î intotwosts, n(î ). Ano Ú Î µwhihisonnttoninthutinll ut-no.astout-nosssoitwithivnut isllut-no-st.ini.7(),ut-no-st Ý Ü.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. Ini.7(),nypthromthroottotrminls mustothrouhithr Ýor Ü.Thror,i Þ Ûrtrt solumninisn Ý Üsrowinis,thnumro istint olumns is xtly two. Th omposition pross ins y noin th BDDnosinthut-no-st. Thisisshownini.8. Th numr o its(vrils) rquir or th noin is ÐÓ Òµ,whr Òisthrinlityothut-no-st. or this xmpl, on it(vril) is suiint. A nw vril,,isintrou.thbddo notin y sustitutin Ýn Üwith thir rsptiv os, s shownini8(). ThisrsultsinthinlAshnhurst omposition, Ü ¼ Ý,whr ÞÛ Þ ¼ Û ¼. Althouh n optiml omposition or th ov untionnounythmthos,itisnotths or nrl, omplx Booln untions. Du to lk o ritrion or oo ut, ut is usully prorm whn th numr o vrils ov th ut is lss

8 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 8 w z y x w w y z x w i. 8. Ashnhurst omposition usin BDD. = zw + z w = x + y thn som ix vlu,. Th pplition o ths mthos r hn rstrit to Look-Up-Tl(LUT)- spgas,with inthnumroinputston PGAlok[26],[27]. Wlivtht,withthhlpo struturl nlysis o BDDs, this typ o omposition n xtn to ompositions lin to iint multi-lvl implmnttions. WrlsowronpprohinwhihBDDis us s n inirt orm to unovr oo ompositions. In [28], sust o sptrl oiints o Booln untion, rprsnt s BDD, is lult. Th BDD 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 Booln untion y nlyzin th strutur o its BDD wsirststuiykrplus[29]tthrlysobdds. Hintrouthonpto-n-omintor 5,whih l to n lri AND/OR omposition. i. 9 illustrts th onpt o - n -omintor. Bsilly, -omintor(-omintor) is no whih lons to vrypth Ô ½ (Ô ¼ ).Thxistno-omintor (-omintor) llows th BDD to ompos into two prts onjuntivly(isjuntivly). -omintor () -omintor i. 9. Exmpl o - n -omintors introu y Krplus. () - omintorlstonlrionjuntivomposition, µ µ. ()-omintorlstonlriisjuntiv omposition,. Both-n-omintorsrspilssoournrlizomintor, 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 BDD struturs. Brto t l.[3] propos mtho whih prorms hirrhil isjuntiv omposition irtly on BDD. This mtho silly nnotts isjuntiv omposition inhrnt in th BDD 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 BDDs with omplmnt s. Stnion t l.[3] propos nrliz otors Booln ivision n toriztion mtho. Givn ivisor, untion n writtn s Ó µ ¼ Ó ¼ µ. Consquntly,Booln ivision is prorm y sttin É Ó µ n Ê ¼ Ó ¼ µ. Thrsultnurthrimprov y rlizintht É, n Ê imply on trsts to h othr. Howvr, u to lk o iint wy to nrt Booln ivisors, th improvmnt o this mtho ovr SIS is mrinl. Nithr o th ovmntion mthos rss nrl omposition o BDDs onto xprssions involvin XOR loi. B. unmntls Bor ivin into th tils o irnt typs o BDD ompositions, lt us irst provi thortil nlysis o two unmntl ompositions, nmly Booln ivision n Booln sutrtion. All othr typs o ompositions n riv rom ths two. Dinition8(BoolnDivision) untion isbooln ivisoro ithrxistsuntion É,llquotint,suh tht É. In[32],Boolnivisionisins É Ê, n isllboolntor. Inouromposition shm,wlwysssum Ê. Toomplywithth trms ivision,wll Boolnivisor,insto tor. Inthisppr,wshllusthtrmsBooln ivision, onjuntiv Booln omposition, n Booln AND omposition, intrhnly. Dinition9(BoolnSutrtion)untion isbooln sutrtoro ithrxistsuntion Ê,llrminr, suhtht Ê. In th squl, w shll us th trms Booln sutrtion, isjuntiv Booln omposition, n Booln OR omposition, intrhnly. Thorm2: [33]untion isboolnivisoro i nonlyi. Proo: I isboolnivisoro,thnthrxists É suhtht É ; É µ.onthothrhn, µ ʵ É. Hr Êisny untion Ê ¼. Exmpl2:Consirtwountions n. Sinthon-sto isovrythto,i.., isboolnivisoro. untion nomposs µ µ.thbddsor untion n rshownini..

9 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 9 Thorm3:Auntion isboolnsutrtoro i nonlyi. Proo:ThprooisulothtoThorm2. Dinition (Co-torin squn) Consir no Ú inbdd.thpthromrootto Únuniqulyin sstovrils,whrhvrilmypprintru oromplmntorm. Suhstovrilsisllotorinsqun. I Úistrminlno,thlistisll trminl o-torin squn. Lt us now stuy th proprtis o Booln untions n É in orr to stisy É. Sin BDD is rphil rprsnttion o squn o Shnnon xpnsions o th Booln untion, th pross n rily monstrt y usin squn o o-torin oprtions. nxpnusinshnnonxpnsion s: Ü ¼ Ü ¼ Ü Ü () É nxpninthsmwy,wotin É Ü ¼ É Ü ¼ ÜÉ Ü µ Ü ¼ Ü ¼ Ü Ü µ Ü ¼ É Ü ¼ Ü ¼ ÜÉ Ü Ü (2) Thn É ithollowintwoonitionsrstisi, É Ü ¼ Ü ¼ Ü ¼ É Ü Ü Ü (3) By inution, th ov onitions n nrliz to ny o-torin squn Û. Tht is, i É Û Û Û (4) istruornyo-torinsqun Û,thn É. Whn Én rrprsntsbdds,tohk whthr É is tru, only trminls nto hktosionition É Û Û Û isstisi.whn Ûistrminlo-torinsqun,oronition Û É Û Û totru, Û n É Û muststisythollowin two onitions, Û ½ µ Û ½ É Û ½ (5) Û ¼ µ Û ¼ É Û Û É Û ¼µ (6) whr*stnsoron tr. Thorm4(Boolnivisoronition) isonjuntivboolnivisoro,iorvrytrminlo-torinsqun Û or ivn vril orrin, Û ½ µ Û ½ Û ¼ µ Û ¼ Proo: Sin Û ½ µ Û ½ornytrminlotorinsqun Û, ÓÒ ÓÒ. Hn isbooln ivisoro. Thorm 4 provis n iint wy to hk whthr Booln untion is ivisor o nothr Booln untion. Aswillomlrinthollowin (7) stions, Thorm 4 provis th thortil ountion or nrliz omintor. In th sm mnnr, th onition or Booln sutrtor n lso ormult. Thorm5(Boolnsutrtor) isisjuntivbooln sutrtor o, i vry trminl o-torin squn Û or ivn vril orrin, Û ½ µ Û ½ Û ¼ µ Û ¼ Proo:ThprooisulothtoThorm4. C. AND/OR Domposition In this stion, irnt typs o BDD ompositions trtin AND/OR loi omposition r prsnt. C. Booln Domposition irst, th most nrl strutur lin to Booln AND/OR omposition is xmin. This strutur is rrr to s nrliz omintor. Dinition (Gnrliz Domintor) Consir ut prtitioninthstobddnosountion into n (V-D).ThportionothBDDiny isopitoorm sprtrph. Inthtrph,n isonntto i ¼ inthoriinlbddo,nitisonntto i ½ inthoriinlbddo. Allthintrnls µrltnlin.thrsultinrphisll nrliz omintor. i shows th onstrution o nrliz omintor. Ini.(),utisprormonthBDD.Thnth portionovthutisopitosprtrph,whih is shown in i. (). Th onstrution is omplt y onntin s o th rph to th orrsponin trminls in th oriinl BDD. Not tht us o th nlins,nrlizomintor isnotbdd.by ssinin th nlin s to irnt onstnt vlu( or), nustoomposbddonjuntivlyor isjuntivly.lt stollnlins. = + () () i.. Gnrtion o Booln ivisor s on nrliz omintor. Th ollowin thorm shows how to otin Booln ivisor n prorm th ivision 6 y rirtin th nlins o toonstntno. Thorm6(Construtiono É, ) Givn nrliz omintor ountion,thboolnivisordisotin WlsorrtoitsBoolnANDomposition (8)

10 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM rom yrirtinnlins o to. Th quotint Éisotinyminimizin withtho-sto s on trst. Proo: AorintoThorm,thristlstonpth Ô ½ pssinthrouhhintrnlo.byrirtin thsintrnlso to,thotinbdd(untion) ovrsllpths Ô ½,i.. ÓÒ ÓÒ. Thror, is Boolnivisororuntion (sthorm2). Th quotint É n vrii y hkin whthr th onition Û É Û Û istruorllpossilpths Ûinth BDD.Rlltht Éisopyo,xptorthswhih orrspontotho-sto. Thror,thon-stononminimiz É, É ÓÒ ÓÒ.Sin ÓÒ ÓÒ, É Û Û Û whnvr Û ½. Byonstrution, Û ¼ µ Û ¼, n É Û issttoon tr,sotht É ÓÒ ÓÒ. Thror, É Û Û Û whn Û ¼. ÉisthquotintothisBooln ivision. Exmpl3:ToillustrtthThorm6,simplxmplis shownini..thbddountion isshown ini.(). irst,utisprormonthbdd.thn, nrliz omintor is onstrut s on th ut. Sin Booln ivision is ntiipt, th nlin s on th nrliz omintor r rirt to onstnt. Th Booln ivisor issilyvluts.thquotint Éo thisivisionnotinrom ysttintho-st ¼ ¼ o son tr. Atrminimiztiono withthison t r, É.Notitht µ µ,n É µ µ. = + () D DC Q D () () i.. A simpl xmpl o Booln ivision. Q = + D = + In th ollowin, mor omplx xmpl is provi. Exmpl 4: A omplt onjuntiv(and) omposition, inluinthonstrutionoquotint É,isshownini.2. Ini.2(),utisprorminthBDD.Ini.2(),th nrliz-omintor is otin y opyin th portion ov thtuttorph.thnboolnivisorisuiltyrirtin llthnlinsothtrphto. ThruBDD o islsoshownini.2(). Asinitinthiur, thisompositionxposs-omintorin,whihwsnot prsntinthoriinlbddo. Thror, nsily omposs µ. Ini.2(),quotint É isotinrom yminimizinuntion usinuntion son tr. Thisrsultsin É µ. As rsult o this pross, th whol untion n ompos s µ µ. Booln sutrtion is th ul s o Booln ivision. Th ollowin is th unmntl thorm or Booln sutrtion. Thorm7(Construtiono, Ê)Givn nrliz omintor ountion,thboolnsutrtor o n otinyrirtinnlins o to.th rminr Êisotinyminimizin usinthon-sto son trst. Proo: AorintoThorm,thristlstonpth Ô ¼ pssinthrouhhintrnlo.byrirtin thsintrnlso to,thbddothrsultinuntion ovrsllpths Ô ¼,i.. Ó Ó ( ÓÒ ÓÒ ).Thror,DisBoolnsutrtororuntion (s Thorm2).ThrstothprooisulothtoThorm6. Durin th pross o inin n optiml Booln AND/OR omposition, ll possil uts shoul xris. Oviously, th numr o possil uts oul vrylrvnormiumsizbdd.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. Dinition2(Vliut)Autisllvliiitontins tlston.othrwis,utisinvli. Thorm 8: Only vli uts l to nontrivil Booln omposition. Proo:ConsirninvliutinthBDD.Byinition,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 Booln ivisor (Boolnsutrtor) ½( ¼).Thsssrshownin i.3()n().nowonsirvliut.sin ½ ¼ µ, som o th trminl s o th nrliz omintor r onnt ¼(½),whilothrs( )ronntto ½(¼).Hn thrsultin n É(Ê)isnontrivil,lintonontrivil omposition. Dinition 3(-Equivlnt Cuts) Two uts r -quivlnt ithyontinthsmsusto ¼ s. Dinition 4(-Equivlnt Cuts) Two uts r -quivlnt ithyontinthsmsusto ½ s. Thorm 9(Distint Cuts) All Booln ivisors otin rom -quivlnt uts r intil. Proo: Consir two uts, n, whih r -quivlnt. In h o th Booln ivisors nrt y thos uts, s ¼ ronntto.allothrsronntto. Hn,othBoolnivisorshvthsmpthsromrootto (on-st)nthsmpthsromrootto(o-st).hn,oth Booln ivisors r intil. This is illustrt in i. 3(),(), whih showin tht ut2nut3lonto-quivlntlss,nhnl to intil Booln ivisors. Thorm (Distint Cuts) All Booln sutrtors otin rom -quivlnt uts r intil. Proo:ThprooissimilrtothtoThorm9.

11 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM ru -omintor D = + + D DC Q DC minimiz Q = + + Q -omintor () Oriinl untion () Gnrliz omintor n Booln ivisor () Minimizin with o-sts in D s on t r i.2. OtinintorormonBDD. p 2 3 x y z 4 () Vrious uts on BDD D D ru () Trivil Booln ivisor nrt rom ut D D ru () Trivil Booln sutrtor nrt rom ut D D ru ru () Booln ivisor nrt rom ut 2 () Booln ivisor nrt rom ut 3 i. 3. Et o ut on th nrtion o Booln ivisor/sutrtor. In onlusion, inin ut n viw s prtitionino ¼ n ½ s,rthrthnprtitionin o BDD nos. Thror, th totl numr o ll possil utsis ½ ¼ µ.anin-pthnlysisobddstrutur 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 Ú, nits-(or-).autontinin must lsoontinsllothr sspnnin 7 rompthromroot to Ú. Proo: Th trnsitiv proprty is urnt y th t tht utnnotrossthsmpthmorthnon. Asshown ini.3(),thr ¼ s, Ü Ýn Þoriint(spn)rom noswhihronthsmpth Ôtono Ú. Thror,ny utrossin Ümustlsoross Ýn Þ. Anissitospnrompthiitisininttonoonth pth.

12 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 2 Th trnsitiv ut proprty rmtilly rss th numropossilutsinbdd.howvr,sinth tulnumrovliutspnsonspiibdd strutur,itisiiulttoivonrtormulorth totl numr o vli uts. In our pproh w limit our ttntion to horizontl uts. Our xprin shows tht horizontl uts work wll on mostbdds. Unrthworsts,thtotlnumro horizontlutsis Î,whrVisthnumrovrils (lvlsobdd).inprti,thtotlnumrovli horizontlutsismuhsmllrthn Î,usmny uts r ithr -quivlnt or -quivlnt. C.2 Alri Domposition Alri omposition is spil s o Booln omposition. Du to th importn o th lri omposition n sinss with whih it n intii on BDD, lri AND/OR struturs r rily intii inpnntly o nrliz omintors. Two si struturs lin to n lri AND/OR ompositionwrounykrplus[29]. Hrwrviwths strutursnshowthtthyrspilssoournrliz omintor. Dinition 5(-Domintor) No Ú Î whih lons tovrypth Ô ½ isll-omintor. It shoul not tht th ov inition pplis only tobddswithoutomplmntsovno Ú.ABDD with-omintorhsnshownini.9(). Thorm 2(Alri AND omposition) Th BDD whih ontins -omintor n lrilly ompos intotwoonjuntivprts,i..,,whrthsupports o n risjoint. Proo: i. 4() shows th strutur o -omintor, in whihno Úlisonllpth ½.Iutisprormirtly ov no Ú, th Booln ivisor nrt rom th nrliz omintor is struturlly intil to th portion o th BDD ovthut.thisisshownini.4().thquotintothis ivisionnotinyrirtinth ¼ stoon t r,whihnthnrirttono Ú. ThnllBDD nosinprt hvthsmtrnsitivhil, Ú,nthwhol prtollpssintono Ú.Thisisshownini.4().Sin thrisnoommonsupporttwn n,thomposition is lri. Dinition 6(-Domintor) No Ú Î whih lons tovrypth Ô ¼ isll-omintor. -omintorisulo-omintor. Anxmplo -omintor is shown in i. 9(). Thorm 3(Alri OR omposition) Th BDD whih ontins -omintor n lrilly ompos into twoisjuntivprts,i..,,whrthsupportso n risjoint. Proo: ThprooissimilrtothtoThorm2. Itis illustrt in i. 5. D. XOR Domposition BDD omposition s on nrliz omintors, sriinthprviousstions,rlison s.itis intrstin to not rtin proprtis o s. Nmly, s provi n rly vlution o Booln untion. orxmpl, thvluountion ( )ntrminwhnithr or quls to ¼(½). BDDsountionsthtrminlyomposo AND/ORloitntohvmny s.onthothr hn, BDDs o untions popult with XORs hv vry worno s.thror,thvluountionwith XORs is trmin y th rltiv vlus o its vrils. orxmpl,thvluountion willonly trminwhnvlusoothvrils n r ivn. It is pprnt tht th omposition whih rlis on swillilonbddwithw s. Inthis stion, th thniqus trtin XOR-typ omposition obddrvlop.inthissthomplmnts 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 thprsnoomplmntsinbddisrlt to XOR omposition. In th squl, w will us XNOR( )instoxorusxnorhsmor strihtorwr rprsnttion on BDDs. D. Alri XNOR Domposition Dinition 7(x-omintor) No Ú Î whih is ontininvrypth Ô isllnx-omintor. x-omintor q y r u x i. 6. Rol o n x-omintor in XNOR omposition. ABDDwithnx-omintorisshownini.6. Not tht th inition o x-omintor implis tht thr must xists t lst on omplmnt ov th x-omintor Ú.OthrwisllthBDDnosov Úwillollpsinto Ú. Thror x-omintors o not xist on BDDs without omplmnt s. Thorm 4(Alri XNOR omposition) Lt Ú nx-omintorothbddountion.thbddo n lrillyomposs Ù,whr isbdd roott Ú,n ÙisthBDDroottthoriinluntion with Úrplyonstnt. Proo: i. 7() shows nri BDD with x-omintor Ú. Byinitionoomplmnts,thBDDo roott Ú nsplitintotwoprts, n ¼,sshownini.7(). ThnthBDDnrprsntsisjuntionotwoprts, sshownini.7().nottht n ¼ rth-omintors inthirrsptivbdds. Byinin ÙtothBDDo y x q r u

13 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 3 D Q v DC v v v () -omintor strutur () Gnrtion o th ivisor D () Gnrtion o quotint Q i. 4. -omintor strutur n its orrsponin omposition. D R v v v v DC () -omintor strutur () Gnrtion o th sutrtor () Gnrtion o rminr i. 5. -omintor strutur n its orrsponin omposition. v u u u u () () () Whilxhustivsrhorllpossiluntions is lrlyprohiitiv,stooonitsor n tt irtly orm BDD strutur, ll nrliz x-omintor, in s ollows. Dinition 8(Gnrliz x-omintor) No Ú Î whihispointtoyoththomplmntnrulrs 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 inwhih Úisrplwith,untion nomposs Ù Ù ¼ ¼ Ù. Exmpl 5: An x-omintor is shown in i. 6. Aorin to Thorm 4, th BDD n lrilly ompos s Ü Ýµ Ù ¼ Ö ¼ Õµ. D.2 Booln XNOR Domposition TholoBoolnXNORompositionountion istoinomposition thtwillminimizth ost o its implmnttion. Usully XNOR omposition is prorm on untion in whih oo AND/OR ompositions r unlikly to oun. Thorm 5(Booln XNOR omposition) or Booln untion,ivnnritrryboolnuntion,thrlwys xistsboolnuntion,suhtht. Proo: Th proo is trivil, usin th ollowin Booln trnsormtion. µ (9) whr isnritrryboolnuntion,n. LtBDDo ontinsnrlizx-omintor.by prormintrnsormtion,thrulrs pointin to rrirtto(us ½), nthomplmntspointinto rrirtto ( ¼ ¼).Inthpross,thtrnsormtionrmovs thxor-rltomplmntspointinto.thxnor or o Booln untion n iintly xtrt y rmovin XOR-rlt omplmnt s rom its BDD. Exmpl6:iur 8shows thbddor iruitrn4-,tstsinthmcncnhmrksuit. Aorinto Dinition 8, thr r two nrliz x-omintors in this BDD,nmly Ü ½ Ü,n Ü. Willustrtth ompositionson.itsbddisshowniniur8(). ThBDDo islsoshowniniur8(). Th BDDo onsistsonx-omintor,nthbddo onsistso-n-omintors. Throrothothm nurthrlrillyompos, rsultinin Ü ½ Ü µ Ü Ü Ü ½ Ü µµ. E. MUX Domposition BDD is rphil rprsnttion o squn o Shnnon xpnsions. Eh no in BDD n viw s simpl multiplxor (MUX). Tkin MUX

14 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 4 x2 nrliz x-omintors x3 x x x4 x x4 = x2 x3 x x4 xists in vrious Booln untions, thy r spilly ommon in rithmti untions. Thy r rquntly ssoit with XNOR omposition. Exmpl7:Showniniur2issimplxmplo untionl MUX omposition. Nos Ù n Ú ovr ll pths Ô. Susquntly, untion n omposs ¼,whr ¼ srvssontrolsinl othmux. () () i. 8. XNOR omposition o untion rn4- omposition rrlss o th spii BDD strutur otn ls to poor multi-lvl Booln xprssions. Simpl MUX omposition w.r.t sinl no is only niil whn th ovrlp twn its two o-tors is lssthnrtinthrshol.thississhownini.9. u v () () v v i.2. ExmplountionlMUXomposition: ¼, ¼. 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 6(untionl MUX Domposition) Consir BDDstrutur,inwhihtwonos, Ùn Ú,ovrllpths Ô.ThBDDnthnomposs ¼,whr isotinyrirtinno Ùto,nno Úto,n n runtionsssoitwithnos Ùn Ú,rsptivly. Proo: ThprooissimilrtothtoThorm4. Th omposition is shown in i. 2. u h v u i.2. untionlmuxomposition. ¼. Similr to th initions o th - n -omintor, this thorm pplis only to BDDs without omplmnt s ov Ù n Ú. Whil th untionl MUX omposition v h. Linr Expnsion o BDDs In this stion, nrliztion o irnt BDD ompositions sri in prvious stions is stui. It will shown tht ll prvious BDD omposition mthos r spil ss o linr xpnsion to prsnt hr. Our ojtiv is not to ovrrul ll spil-s BDD 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 Booln omposition. i. 22()shows nri BDD. Eh rprsnts n ritrry loi untion, inluin onstnt untions n. Any BDD oul rprsnt in this wy without loss o nrlity. Lt us xmin th omposition o suh strutur BDD into st o isjuntiv omponnt BDDs ( ½ ), shown in i.22().ehomponntbdd onsistsooiint BDD nuntionbdd. Notthtthrooto huntionbdd plysrolo-omintorinth rsptiv omponnt BDD. Thror, h omponnt BDD nurthromposorintoth- omintor strutur. Th inl omposition is shown in i.22(). Now lt us stuy th proprtis o oiint BDDs ( ½ ). Th rltiontwn thos oiints r shown in i. 23. Sin ll oiint BDDs r nrtromthsmbdd,nironlyinthir trminls, ll oiint BDDs r rphilly isomorphi. Aorin to th prinipl o APPLY oprtion[5], th Booln oprtions twn thos oiint BDDs only tk pl t th trminls. Thror, th union o ll oiint BDDs is qul to, whih is shown in

15 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 5 2 k 2 k 2 k () A nri BDD () Linr xpnsion o th BDD 2 k 2 k () Domposition o ll omponnts usin -omintor i.22. LinrxpnsionoBDD i. 23(). Similrly, th intrstion twn ny two irnt oiint BDDs is qul to, whih is shown in i. 23(). Mthmtilly th ov nlysis n ormult s ollows. Thorm 7(Linr Expnsion) A Booln untion n xpn w.r.t. n orthonorml oiint st, ½,sollows: ½ È whr ½ ½n ¼. Proo: Any Booln untion n rprsnt s BDD withstruturshownini22.thisiurni.23provi th proo. W not tht our linr xpnsion thory souns xtly lik th Dinition 2(orthonorml xpnsion). Howvr, th wy in whih th two xpnsions rrriout is irnt. Whn Booln 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 BDD, us th oiint BDDs nuntionbdds rrprsntxpliitly ybddstrutur. Thonlythinthtnsto onistoiuroutwhihstooiintsshoul us or th omposition. Similrly, BDD strutur provis lots o hints or this typ o omposition; som struturl nlysis o BDD is rquir or this purpos. Th tivnss o linr xpnsion n lso rily stimt y th nlysis o th BDD strutur. In summry, Thorm 7 provis urthr lxiility to ompos n ritrry BDD. Th ppliility o thisthormrlisonininbddstruturtowhih 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 BDDs 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 BDD omposition shm. V. LOGIC SYNTHESIS BASED ON BDD DECOMPOSITION -BDDLOPT In this stion, implmnttion tils o th loi optimiztion prorm, BDDlopt, whih is s on our BDD omposition thory, r prsnt. Alorithmi nlysis o prours in th propos loi synthsis low is lso provi. Itwillshownthtllnssryproursintypil loi optimiztion low n implmnt throuh sris o BDD mnipultions n ompositions. or xmpl, Booln simpliition n iintly rri out throuh BDD vril rorrin; toriztion n on throuh rursiv BDD ompositions; n loi shrin n iintly tt on th inl torin trs. A. Synthsis low ThsynthsisloworBDDloptisoutlinini.24. Th low onsists o two mjor prts, BDD omposition n torin tr prossin. irst, th lol BDDs(s Stion II-C) r onstrut or th Booln ntwork. Thn, th lol BDDs r sumitt to th omposition nin or loi omposition. Alon with th BDD omposition, st o torin trs r onstrut to ror th omposition. In th pross o BDD omposition, lr BDD is rursivly ompos into sml prts. Th omposition pross stops whn

16 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 6 2 k () Sum o ll oiints quls. i j i j () Intrstion twn ny two irnt oiints quls. i. 23. Coiint proprtis BDD Domposition Enin Construt On No on th torin Tr N Booln Ntwork Construt Glol BDDs BDD hs on no? Y torin Tr Prossin Synthsis Rsult Prsnttion Thnoloy Mppin i. 24. Synthsis low o BDDlopt BDD hs on no. inlly, n importnt prour, shrin xtrtion, tks pl in th torin tr prossin phs. Bus 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. B. BDD stor/lo Mhnism In this stion, BDD mnipultion thniqu, whih is ruil to prormin loi simpliition in BDDlopt, is xplin. In our BDD-s loi optimiztion shm, BDD vril rorrin lorithm srvs s n impliit loi simpliition. It shoul mphsiz tht, in typil BDD pk, vrils r rorr with rspt to BDDmnr,nnotw.r.t. spiibdd.hn,i thrismorthn on BDDin th mnr, vril rorrin my not rsult in th sir simpliition orspiibdd.inorrtohivmximumloi simpliition o Booln untion(bdd), ll othr BDDs must r rom this BDD mnr or prormin vril rorrin. Howvr, thos r BDDs must prsntinthbddmnrwhnthyrn or omposition t ltr tim. Thror, n iint stor/lo mhnism must vlop. AnivwyostorinBDDistoumpitintoSOP orm. ThvntoSOPormisthtthBDDn ronstrut unr vril orr whih is irnt romthorrinwhihthbddisstor. Thisors som lxiility or th implmnttion. Howvr, sin thnumrosoptrmsobddnxponntilin thnumrobddnos,storinbddsinsopormis not sil solution. A nw t strutur, Pool, hs n vis to prorm BDD stor/lo oprtions. Bsilly, Pool isdagwhihisrphillyisomorphitothbdd it rprsnts. A BDD is opi to Pool or it is rrom th BDD mnr. Th BDD n ronstrutltrypplyinnit 8 oprtion Òtims, whr Ò is th numr o BDD nos. Sin n it oprtion tks onstnt tim, th ovrll omplxity o our BDD stor/lo lorithm is Ç Òµ. Th isvnt o PoolisthtthvrilorrothBDDmnr intowhihbddislomustthsmsthorr inwhihbddisstor. orinbddmnrto rtin vril orr oul rsult in n xponntil inrsinthbddsizithmnrisnotmpty. Anitisshortori-thn-ls;itisins Ø Ü µ Ü Ü ¼

17 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 7 Howvr, in our pplition, whn BDD is lo (uilt), th BDD mnr is lwys mpty. Anothr importnt tur o our Pool mhnism is to llow th vril sustitution urin th pross o BDD ronstrution. This n omplish sily y moiyinthitoprtors Ø Å Üµ µ,whil Å is th mppin o vrils. This tur plys ruil rol in our iint itrtiv limint prim(stion VI- C). C. Th BDD Domposition Enin ShowninAlorithmisthminprourorth BDD omposition. To mk th BDD mnr ln, llbddsrstorinth Poolorm. ABDDis lo into th BDD mnr or it is ompos. Th stor/lo pross is rliz y untion storb n lob. AtrBDD hsnonstrutin th BDD mnr, it is ompos y omposb. Thompositionrsultsrprsnts,whr stnsorboolnoprtor,suhsand,or,xor orxnor.thompositionisstorinthormo torin tr, isuss in Stion V-D. Th intrmit BDDso n rthnstorinth Poolorm nnquuithyhvmorthnonno. Th omposition pross is itrt until th quu is mpty. Pool = storb(); Enquu(Q, Pool); whil(pool = Dquu(Q)) { = lob(pool); (, h, op) = omposb(); onstrut on no on torin tr; i (!= sinl no) { Pool = storb(); Enquu(Q, Pool) } i (h!= sinl no) { hpool = storb(h); Enquu(Q, hpool); } } rturn (torin tr); Alorithm : Min BDD omposition low Th min BDD omposition nin, omposb, issrhprossorthmostiintbddomposition, rom mor iint(lri) to lss iint (Booln). 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 BDD is ompos usin otor w.r.t. th top vril. In prti, th lststpisrrlyrh.itisputhrtonsurthbdd willomposwhnllothrttmptsil. A BDD omposition pross ins with th BDD struturl sn, in whih th struturl inormtion o BDDisotin. Thinormtionisussuin 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 omplxityo Snis Ç Ò Î µ,whr Òisthnumr obddnos, Î isthnumrovrilsinthbdd. C. Simpl Domintor All thr simpl omintors (-, - n x-omintor) shrsimilrpttrn,i..,thrisnointowhih ll intrnl s onvr. Bs on this osrvtion, iint lorithms n sin to unvil ll simpl omintor struturs. In t, th BDD sn prour Sn is vis to in out th struturl inormtion o BDD. Th struturs o simpl omintors r lry no in th t ollt y Sn. Th omplxityothisuntionis Ç Î µ. 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 BDD is rturn. This hlps to hiv mor ln omposition, whih is ruil to th ly minimiztion. C.2 Gnrliz Domintors n Gnrliz x-domintors I n lri omposition os not xist or Booln untion, Booln omposition will prorm. Th BDD struturs lin to Booln 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 BDD minimiztion w.r.t. on t r. Thror, th omposition rsult pns on th iiny o th BDD on t r minimiztion lorithms. To rry out ths ompositions, ll possil Booln ompositions r xmin lvl y lvl. BDD sn inormtion is rquir or th pplition o vrious iltrs. On h lvl, two mjor stps, nrliz omintor nrtion n BDD minimiztion w.r.t. on t r, r involv in sinl omposition. Th nrtion onrlizomintorisprossoopyinthbdd strutur ov th ut. Th uppr oun or this oprtionis Ç Òµ.Thuntionustolult (or ¼ ) is s on RESTRICT oprtor[34] whos omplxity is Ç µ(or Ç ¼ µ).thupprounorrestrict is Ç Ò µ. Thror,upprounoruntion omp- GnrlizDomintoris Î Ò µ. D. Constrution n Prossin o torin Trs AtorintriswytororBDDomposition pross. or xmpl, i Booln untion is

18 YANG AND CIESIELSKI: BDD-BASED LOGIC OPTIMIZATION SYSTEM 8 omposinto,thnnwno,withoprtor + ntwosilins, n,willrttoror this omposition. A torin tr will kp rowin until th BDD 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, BDDs r onstrut or ll torin trs in ottomupshion. ThnoniityproprtyoBDDisusto 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: ) AND/OR-intnsiv untions, n 2) XOR-intnsiv loi (rithmti untions). Th litrl ount or ompositions nrt y BDDlopt 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. Both trs SIS mppr n Booln mthin-s rs[35] r us. rs is s on Booln mthin rthr thn tr mthin. or this rson th XOR ompositions oun y BDDlopt r likly to prsrv. Th rsults or AND/OR intnsiv iruits r shown in Tl I. On vr, BDDlopt uss slihtly wr ts thnsis,nmorrthnsis.thslihtinrsin risutothhihrostoxortsimplmnt in CMOS. On vr, th inl synthsis rsults usin BDDloptnSISon this lssountions rlmost th sm. Whil nr optiml rsults r otin y oth SIS n BDDlopt, ut BDDlopt out-prorms SIS rmtilly in CPU tim. Howvr, or th lss o rithmti untions n XOR-intnsiv loi, shown in Tl II, BDDlopt 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 ytsitl[36]rlsolistoromprisonpurpos. OnnsthtthprormnoBDDloptintrms o th numr o ts is omprl to tht o Tsi tl.[36]. ItshoulnotthtmnyXORsinth ntlist synthsiz y BDDlopt r lost tr thnoloy mppin.asinitinolumnxorsintlii,only ± XORs r prsrv in thnoloy mppin. VI. BDD-BASED LOGIC SYNTHESIS SYSTEM- BDS 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 prolmitsl. Inours,thsizoBDDshoul proportionl to th siz o iruit(whih is ommonly msur y th numr o loi ts). Howvr, th siz o lol BDDs or ivn Booln ntwork is ompltly unpritl. It stronly pns on th typothiruit,rthrthnonthtotlnumro ts. Rprsntin th ntir Booln ntwork y lol BDDs uss srious omputtionl prolms. Thror, propr prtitionin o th Booln ntwork is rquir prior to prormin th BDD omposition. Tl III showsthomprisonothsizololbddsnlol BDDs(ininStionII-C). Itnounthtth sizololbddsoulsmuhstwoorrso mnitu lrr thn lol BDDs. 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 Booln ntwork, its multilvl strutur shoul prsrv s muh s possil. ThnumroSOPtrmsoultoolrorthloi optimiztion lorithms i th ntir Booln ntwork is ollps into two-lvl orms. rom this point o viw, th ntwork prtitionin y BDD-s loi synthsis is similr to th on y tritionl multilvl loi synthsis. Ciruits Glol BDDs Lol BDDs C C C C C C C C C pir rot TABLE III COMPARISONONUMBEROBDDNODESORGLOBALANDLOCAL CONSTRUCTION In this stion, nw loi synthsis systm, BDS, whih hs th pility to optimiz ritrrily lr iruits, is prsnt. A. Synthsis low Currnt multi-lvl loi synthsis low xmplii ysishsrwnromovrtwntyyrsointnsiv rsrh. W liv it hs th pility to hnl vrylriruitsnitosrspthssnoloi synthsis in nrl. Thror, BDS opts th nrl synthsis low o SIS. i. 26 omprs th synthsis low o SIS n BDS. Th similrity twn thm is ovious. Th unmntl irn twn SIS n BDSisthwyinwhihhsystmrprsntsBooln