Boolean Function Representation Based on Disjoint-Support Decompositions.

Transcription

1 Booln Funtion Rprsnttion Bs on Disjoint-Support Dompositions. Vlri Brto n Murizio Dmini Diprtimnto i Elttroni Informti Univrsit i Pov, Vi Grnigo 6/A, 353 Pov, ITALY Astrt Th Multi-Lvl Domposition Digrms (s) of this ppr r nonil rprsnttion of Booln funtions pliiting isjoint-support ompositions. s llow th rution of mmory ouption with rspt to tritionl ROBDDs y omposing logi funtions rursivly into simplr - n mor shrl - loks. Th rprsnttion is lss snsitiv to vril orring, n us of this proprty, nlysis of th grphs llows t tims th intifition of ttr vril orrings. Th intifition of mor trminl ss y Booln lgr thniqus mks it possil to ompnst th itionl - smll- CPU tim rquir to intify th isjoint-support omposition. W pt th proprtis of s to usful in svrl ontts, most notly logi synthsis, thnology mpping, n squntil hrwr vrifition. Introution Ru, Orr Binry Dision Digrms (ROBDDs) [] r proly th most powrful t strutur known so fr for th mnipultion of lrg logi funtions, n for this rson thy hv om prvsiv in logi synthsis n vrifition nvironmnts [2, 3, 4, 5]. Ongoing rsrh is ttmpting to tn thir ppliility to othr omins, suh s wor-lvl vrifition [6], th solution of grph prolms n intgr-linr progrmming [7, 8]. Still, som ky inffiinis (n ponntil lowup for som lsss of funtions, th unpritility of th ROBDD siz n shp with rspt to th vril orring hosn, t...) motivt n inrsing rsrh tivity in this r, inluing: Effiint implmnttions [9, ], vlopmnt of orring huristis [, 2, 3], n ltrntiv rprsnttions ltogthr [4, 5, 6, 7, 8, 6, 9]. ROBDDs r losly rlt to trministi utomt: input its r vlut squntilly, on t tim, long th grph [9]. In this ppr, w to th si ROBDD rprsnttion th pility of omposing funtion into n ritrry, multipl-lvl tr of isjoint-support su-funtions. Unlik ROBDDs, nos rprsnt not only two-input MUXs, ut lso unlimit-fnin OR / AND (or NAND-only, NOR-only) trs of gts. Th novl rprsnttion rtins most of th proprtis of ROBDDs, nmly, noniity, irt-yli grph Rsrh prtilly fun y th EC ESPRIT III Bsi Rsrh Progrmm ontrt No. 972 (Projt GEPPCOM) n y CNR grnt # CT7. Currntly with Synopsys, In. 7 Est Milfil Ro, Mtn.Viw, CA 9443 strutur, rursiv onstrution thniqu, n onstnttim omplmnttion. Tr omposition rings svrl vntgs. First, th rprsnttion is signifintly lss orr-snsitiv thn ROBDDs. For instn, fully omposl funtions r rprsnt y miniml tr iruits, of siz linr in th numr of vrils, rgrlss of vril orring. whil ROBDD siz n rng from linr to ponntil. Morovr, for this lss of funtions, som iffiult prolms (lik, Booln NPN mthing [2]) n solv in linr tim y tr mthing thniqus. Son, th nw rprsnttion is usully mor ompt thn ROBDDs, us funtions r ompos in simplr, mor shrl loks. Mor intrstingly, th nw rprsnttion givs us mor systmti n t insight on th rol of th input vrils of logi funtion, othrwis frr to spil-purpos huristis suh s ynmi rorring. Evntully, w show tht th itionl CPU tim for omposition is provly smll. Morovr, th nw rprsnttion llows us to intify mor trminl ss, n thrfor to otin fstr n shllow rursions. For instn, th omputtion of f + g n rri out in onstnt tim if f n g shr no vrils, n th siz of th rsult is jfj + jgj +, rgrlss of vril orr. Othr simplifitions ris from th rognition of ommon trms in th omposition of th oprns. In prti, w foun th CPU tim lwys to los n oftn ttr thn tht of rfrn ROBDD pkgs. For rsons of sp, thorm proofs r not inlu. Thy will vill upon rqust. 2 Disjoint support omposition W onsir th omposition of funtions into th NOR (NAND, OR, AND) of isjoint-supportsufuntions, whnvr possil. This notion will l to rursiv (.g. tr) omposition styl n to th finition of s. Dfinition. Lt f : B n! B not Booln funtion of n vrils n. W sy tht f pns on i i is not intilly. W ll support of f (init y S(f)) th st of Booln vrils f pns on. A st of non-onstnt funtions ff f k g, k, with rsptiv supports S(f i ) is ll isjoint-support NOR omposition of f if: f + + f k = f; S(f i ) \ S(f j )= i 6= j () A isjoint support NOR omposition is miml if no funtion f i is furthr omposl in th OR of othr funtions with isjoint support. W fin isjoint support OR, AND, NAND ompositions in similr fshion. W init y D NOR (f) ny suh miml omposition. 2

2 ) ) Figur. A rursivly omposl funtion. Empl. Th funtion f =( + )( + ) hs th following miml isjoint-support ompositions: AND: ff =( + ); f 2 =( + )g; NOR: ff =( + ) ; f 2 =( + ) g; NAND n OR: ffg. 2 In th rst of th ppr, miml isjoint-support ompositions r rfrr to s ompositions, for short. Morovr, w fous only on NOR omposition, s th rsults for th othr ompositions n otin rily y stnr Booln lgr. By omposing rursivly logi funtion, w otin NOR-tr rprsnttion of F : Empl 2. Th funtion F =( + )( + ) is rursivly NOR omposl. From th first omposition w otin f = ( + ) n f 2 =[ +( + ) ]. Ths funtions r thn gin omposl until rhing th input vrils, s in Fig. (). 2 Dfinition 2. A tr omposition of logi funtion f is omposition of f into NOR-only tr of sufuntions, whr th funtions t th inputs of h NOR r mimlly ompos. W inity TD NOR th ompositiontr. Similrly w n fin TD NAND n TD AND/OR. 2 Thorm () low stts rlvnt intuitiv rsult: Thorm. For givn funtion f, ) thr is uniqu D NOR ; n 2) thr is uniqu TD NOR. 2 3 Multi-Lvl Domposition Digrms W ploit tr ompositions to riv hyri rprsnttion styl. Th mol is s on pplying tr omposition whnvr possil, n thn Shnnon pnsion until rhing primry input vrils or thir omplmnts. Empl 3. Th funtion f = ( + + f ) hs TD NOR s in Fig. (2.). Not tht w oul not ompos + us of th isjoint support onstrint. Applying Shnnon pnsion, in Fig. (2.) w otin TD NOR for h input of MUX. 2 Dfinition 3. A is irt yli grph, with lf vrtis ll y Booln onstnt or vril n ) ) + f f Figur 2. ) Tr omposition of th funtion in Empl (3). ) Th sm funtion with th ition of BDD nos. Figur 3. Son rution rul. ) Mu vrtis. ) NOR vrtis two kins of intrnl vrtis: NOR vrtis hv nonmpty st of outgoing gs, h pointing to ; MUX vrtis hv two outgoing gs, ll n n h pointing to. MUX vrtis r ll y Booln vril. A fins rursivly logi funtion with th following ruls: A trminl vrt ll y vril or onstnt nots th funtion. A MUX m ll y fins: F m = F (m) +F (m) (2) A NOR vrt n with k outgoing gs fins th funtion: F n = f + + f k (3) whr f i, i = ::: k is th funtion fin y th point y g i. 2 In, whil MUX vrtis orrspon to ROBDD nos, NOR vrtis init funtion omposition. Just lik ROBDDs, w impos rution n orring ruls to otin mor ompt n nonil rprsnttion: Thr r no two intil sugrphs; For h vrt, no two pointrs point to th sm ; Eh pth from root to trminl must trvrs susqunt MUX nos in rspt of th vril orring n h vril is vlut t most on on h pth. Th of funtion mths multi-lvl logi iruit in th ovious wy. In th susqunt rwings, irls rprsnt MUXs, whil rrys of squrs rprsnt NORs. It is worth noting tht, unlik ROBBDs, son rution rul rs iffrnt onsquns on th two kins of intrnl vrtis. As skth in Fig. (3), rution of NOR vrt os not us its ltion. In ition to ROBDD-lik ruls, in orr to grnt noniity w must impos omposition ruls: th funtions point y NOR vrt must rprsnt miml omposition. funtion is rprsnt y MUX iff it is not omposl, nor its omplmnt. Th following rsult is irt onsqun of th noniity of tr ompositions n rution ruls: Thorm 2. Ru Orr Dompos s r nonil. 2 Th following rsults on D NOR s r usful in th onstrution of th or prours: Thorm 3. Suppos ff f k g is D of som funtion. Thn, y rsing lmnts from th st, th nw st is lso D. 2

3 ; NOR (ml op, ml op2, int i) f if (trminl s) rturn (trminl vlu); 2 D(op) = D(op) \ D(op2); 3 D(op) = D(op) n D(op); 4 D(op2) = D(op2) n D(op); 5 if (trminl s) rturn (D(op) [ trminl vlu); 6 rs = omp lookup(op, op2); 7 if (rs!= NULL) rturn (D(op) [ rs); 8 = top vr(op, op2); 9 l=nor (op., op2., i-);ri=nor (op., op2., i-); rs = ml fin (l, ri, ); omp insrt (op, op2, rs); 2 rturn (D(op) [ rs); g Figur 4. Psuoo of NOR() Thorm 4. If D NOR (f ) =ff f k g[fp p h g n D NOR (g) =fg g l g[fp p h g, whr g i 6= f j i = l j = k, thn:. D NOR (f + g) =(fp p h g[ [ D NOR ([(f + ::: + f k ) +(g + ::: + g l ) ] )) 2. Lt not vril, n suppos f = ( + g). Thn, D NOR (f )=fg[d NOR (g ) 3. Lt not vril not in th support of f or g. Thn: D NOR ( f + g) =fp p h g[ D NOR ([ (f + ::: + f k ) + (g + + g l ) ] ) 2 4 mnipultion routins W tst two istint implmnttions of s. In th first implmnttion, vrtis r rliz uniformly with n-tupls, th first lmnt ing n intgr, ll th othrs ing pointrs to othr s. In th first lmnt w no th typ of no (i.., MUX or NOR vrt), th numr of lmnts in th n-tupl (2 for MUX vrtis) n th top vril of th funtion rprsnt. In th son implmnttion, NOR vrtis r implmnt y link lists. Although th mmory ouption of singl list is twi thn tht of th orrsponing rry, this implmnttion llows th shring of list lmnts. In prti, w foun littl iffrn in trms of CPU tim or mmory ouption twn th two lists. In ithr s, w mintin th strutur in strong nonil form, (i.., no two opis of th sm grph ist), y th usul hshing. W implmnt Booln oprtion routins. Fig. (4) rports th psuo-o for th NOR of two funtions. NOR is invok y th ntlist prsr. For h ll, th prsr knows th support of th two funtions n it trmins n uppr oun i on th rursion pth, nmly, th pth of th lst vril in ommon twn op, op2. Th rursion is strutur s follows. First, trminl ss r intifi. Som trminl ss r inu y th omposition. Thy r rport in lins 4-6 of Tl (). In lin 4, w rogniz whthr op, op2 r of typ op = f + g::; op2 = f + h + :::: In this s, op', op2' ontin f f n w rturn. This is mor gnrl thn just hking op = op2'. W lso hk whthr on oprn pprs s omponnt of th othr. Sin snning th two lists t h rursion stp is pnsiv, only th first list lmnts r tully ompr, ftr th two top vrils hv n trmin. trminl s rturn vlu op=orop2= 2 op = n op2 = op2, op 3 op = op2 op DSD(op'); 2 DSD(op2') 5 6 S(op) \ S(op2) = op = f ( : : :), op2 = fop, op2g ff, g Tl. Trminl ss for NOR() In lin 5, w tt if op, op2 hv isjoint support. In this s, w rt n rturn 2-input NOR, with inputs pointing to op, op2, rsptivly. Sin this rus to tst to th input prmtr i, it tks onstnt tim, rgrlss of th vril orring. In lin 6, w onstrut f (in tim linr to th siz of f, in th worst s) n rturn NOR gt rprsnting (f + ). Rows 2, 3 n 4 of NOR r th pplition of Thorm 4, s. D(op) inits th st of lmnts of th omposition of op. In NOR vrt op, it is th st of ll outgoing pointrs (n inits st iffrn oprtion). W sk ommon lmnts in th oprns n rmov thm from th rursion. This rmovl n rsult in fstr ution, s th nw oprns hv fwr vrils. In prti, only th first lmnt of th two lists is hk, for rsons of sp. Aftr this rmovl, w hk th oprtion to s if it rus gin to trminl s. A omput tl mintins prtil rsults. It is look up in Lin 6. Th rmovl of ommon sufuntions rus tl ovrwrits, s w ploit th ntry F + G = H for rtriving rsults of vry oprtion of typ Ff + Gf = Hf. If th srh fils, rursion strts. ; vltop (ml op, ooln vlu) f if (op is MUX no) rturn (op.vlu); 2 i = lmnt of op suh tht op.topvr = op.i.topvr; 3 opr = vltop(op.i, vlu); 4 D(op) = D(op) n op.i; 5 D(rs) = D(opr) [ D(op); 6 rturn (rs); g Figur 5. Psuoo of vltop() Unlik ROBDDs, oftoring my nontrivil. Prour vltop(f, vlu) rturns th oftor f =vlu ssuming is th top vril of f. Aftr rursion, ml fin() rts from top vr n its oftors. W now nlyz in mor til oftoring n rtion. Figur 6. An mpl of vltop() pplition Th psuo-o of vltop is in Fig. (5), n Fig. (6) illustrts its oprtion. vltop() rursivly gos own

4 ; ml fin(ml lft, ml right, top vr ) f if (lft == right) rturn (lft); 2 if (right == ) f 3 nw vrt = fin or rt(,, ); 4 D(rs) = nw vrt [ D(right); 5 rturn(rs); g 6 if (right == ) f similrly g 7 if (lft == or lft == ) f symmtri s g 8 D(op) = D(lft) \ D(right); 9 if ( D(op) = ) rturn( fin or rt(lft, right, ) ); D(lft) = D(lft) n D(op); D(right) = D(right) n D(op); 2 nw vrt = ml fin(lft, right, ); 3 D(rs) = D(op) [ nw vrt ; 4 rturn(rs); g Figur 7. Psuoo of ml fin() th omposition tr (Lin 3) until it rhs th MUX no ll with th top vril of th (Lin ). Sin h gt ontins th inition of th ritil input with th top vril, only on pth of th tr is trvrs. Rturning from rursion, it tks th oftor of th rh MUX n sustituts NOR vrtis with nwly gnrt ons, (ott in Fig. (6.)) whil mintining noniity. In prti, sin oftn th grph of f =vlu ws rt y prvious omputtion, nw gts r rrly gnrt. Morovr, th rursion is typilly vry shllow (on or two lvls in most nhmrk mpls). ) ml H ml I ml J ml H ml I ml J Figur 9. Intition of D uring trvrsl - gnrl s To this rgr, w osrv tht is oun y th numr of vrils n it is rthr smll in prti. Constnt-tim omplmnttion Complmnt gs (i..not gts) in ROBDDs llow us to rprsnt f n f with th sm strutur, n onstnt-tim hking of qulity f = g sps up ution. NOT gts giv ris to noniity prolms, s on my hv rprsnttion of g s NOT(f) n nothr rprsnttion root t ROBDD no. this prolm is solv in ROBDDs y pplying pproprit ruls whn rturning from rursion [9]. Tht pproh is tn to s with th hlp of th following rsult: Thorm 5. If logi funtion F is NOR-omposl, thn its omplmnt F is not. 2 From Thorm (5), w group funtions in thr lsss: omposl, with omposl omplmnt, n not omposl. Our gol is to lwys rprsnt funtions of th son lss s th NOT of funtion of th first lss. From Thorm (4), howvr, if F is in th son lss, thn its oftors must in tht lss s wll. Hn, th sitution prior to th ll to ml fin must s in Fig. (), n th stnr omplmnt g rution ruls pply. ) Figur 8. Intition of D uring trvrsl - trminl ss Prour ml fin() is rsponsil for isovring ompositions. Its psuo-o is in Fig. (7). It uils isovring vry possil ommon trm from th two oftors. It prforms two istint oprtions. First, it onsirs ss (rows 2 to 7) in whih on of th two oftors is onstnt. For instn, rows 2 to 5 min th s right =, i.., th funtion to gnrt is f = lft =( + l + + l n ), whr l i r th omponnts of lft. Fig. (8) shows ths trminl ss. Lins 8-3 l with th gnrl s. Common lmnts twn lft n right r ftor out (Fig. (9)). This pplis s 2 of Thorm 4. As mntion, vltop() n ml fin() rpl oftoring n fin or rt() in ROBDDs. Whil ths oprtions r onstnt-tim in ROBDDs, thy my tk O() tim in s, whr nots th tr pth. Figur. Equivlnt s 5 s vs. ROBDDs In this stion w prsnt som omprisons in rprsnting funtions with s n ROBDDs. Eponntil growth s r lss snsitiv to vril orring. Consir th funtions: F n =( + 2 )( ) ( 2n; + 2n ) (4) An impropr orring of th vrils (for mpl, pling th o-ll vrils up top) rsults in ROBDD for F n with ovr 2 n nos []. Morovr, in spit of th simpliity of th funtion, most vril orringsfor F n n prov. Th of th funtion, inst, orrspons to th nturl, 2-lvl NOR rliztion with n+ NOR gts, n is of siz linr in th numr of vrils, rgrlss of th vril orr.

5 ) A B A A ) A B A B Figur. ) ROBDD strutur for th funtion of Empl (4). ) for th sm funtion. Empl 4. Consir th funtion f =( + A)( + B) + ( + )(A + B), with n orring of vrils pling on top. Sin f = 6= f =, ny ROBDD hs th spt shown in Fig. (.). In gnrl, w my think of s whr th two oftors look lik funtion f n of Eq. (4), ut with iffrnt omintion of prouts. Any orring of A B whih optimizs on rnh is oun to su-optiml for th othr rnh of th ROBDD. Fig. (.) illustrts th for th sm funtion. Both rnhs r utomtilly ompos optimlly. 2 Shring of logi. Domposition mks it possil to shr loks of logi tht oul not shr with ROBDDs: Empl 5. Fig. (2.) shows ROBDDs of funtions: F = ( +y )(p+q), G =( +y )(+), H =(p+q)(+). If w hv to rprsnt thos thr simultnously, whihvr orr w hoos, w n shr t most two sugrphs, ithr ( + ) or (p + q) or ( + y ). This is us ROBDDs rprsnt th AND of two isjoint-support funtions putting on ov th othr. Th rprsnttion, inst, n ploit miml shring. 2. F q p y G y y ) n shr ) H q p F G H Figur 2. ) ROBDDs for Empl (5). ) quivlnt s 5. Eprimntl rsults W ompr s ginst ROBDDs on numr of nhmrk iruits. Bnhmrks r ivi in thr stions: multi-lvl iruits, two-lvl n th omintionl prt of synhronous iruits [2]. For th first st of tsts, w us th Brkly vril orring [3], n no ynmi rorring took pl in ithr pkg. W ssum ron implmnttions, in whih in prtiulr h ROBDD no tks thr mhin wors. Morovr ROBDDs hv omplmnt gs. For vrtis w ssum n implmnttion whr h no onsists of n rry. Th first p q lmnt stors no informtion, whil othrs r pointrs. Complmnt gs r us for NOT gts. CPU-tim ws tkn on HP Vtr 5/33 with 48Myts of RAM. From Tl (2), s turn out to mor ompt on vrg of 8%. Som nhmrks giv prtiulrly goo rsults. For mpl, omp n pir r ompos vry fftivly. For omposl funtions, s oftn rsult lso in ttr CPU tim, us trm shring n us fftivly. Th lrgst nhmrks, howvr, rsistnt to omposition, n in ths ss s rsult in lrgr CPU tim pnitur without signifint mmory sving. W implmnt ynmi rorring in our mol with sifting-s lgorithm [3]. Ovr ROBDDs, w hv th vntg to know mor out goo vril orr irtly from th t strutur. In Tl (2) w mk omprisons using for h nhmrk th orr givn y our sifting (intrstingly, th finl orring iffrs from tht givn for ROBDDs.) Vril orring took pl only t th n of ution. Rsults show tht, ftr sifting, s improv slightly furthr ovr ROBDDs. This is us uring sifting w ploit our ttr knowlg of th funtion s strutur n n voi to go through orrings tht giv smll vntg ut lok furthr improvmnts. 6 Conlusions n futur work s hv prov thmslvs ffiint in mking pliit th Ds of logi funtions. This proprty llows us to rh mor ompt, flil n roust grph-s rprsnttion. Morovr, this rprsnttion is mor informtiv on th rol of th support vrils of funtion. W pt ths proprtis to usful in ivrs pplitions, most notly thnology mpping for omintionl iruits n spilly Booln mthing /rhility nlysis for vrifition / ATPG in squntil iruits, whr th ility of omposing funtions in simplr loks is usful for rwing implitions mong nt-stt funtions. Rfrns [] R. E. Brynt. Grph-s lgorithms for ooln funtion mnipultion. IEEE Trns. on Computrs, 35(8):677 69, August 986. [2] O. Court n J.C. Mr. A unifi frmwork for th forml vrifition of squntil iruits. In Pro. ICCAD, pgs 26 29, Novmr 99. [3] S. Mlik, A. R. Wng, R. K. Bryton, n A. Sngiovnni- Vinntlli. Logi vrifition using inry ision igrms in logi synthsis nvironmnt. In Pro. ICCAD, pgs 6 9, Novmr 988. [4] Y. Mtsung n M. Fujit. Multi-lvl logi optimiztion using inry ision igrms. In Pro. ICCAD, pgs , Novmr 989. [5] H. Touti, H. Svoj, B. Lin, R.K. Bryton, n A. Sngiovnni-Vinntlli. Impliit stt numrtion of finit stt mhins using BDD s. In Pro. ICCAD, pgs 3 33, Novmr 99. [6] R. E. Brynt. Binry ision igrms n yon: nling thnologis for forml vrifition. In Pro. ICCAD, pgs , 995.

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