Dynamic Minimization of Sentential Decision Diagrams

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1 Dnmi Minimiztion of Sententil Deision Digrms Arthur Choi n Ann Drwihe Computer Siene Deprtment Universit of Cliforni, Los Angeles {hoi,rwihe}@s.ul.eu Astrt The Sententil Deision Digrm (SDD) is reentl propose representtion of Boolen funtions, ontining Orere Binr Deision Digrms (OBDDs) s istinguishe sulss. While OBDDs re hrterize totl vrile orers, SDDs re hrterize more generll vtrees. As oth OBDDs n SDDs hve nonil representtions, serhing for OBDDs n SDDs of miniml size simplifies to serhing for vrile orers n vtrees, respetivel. For OBDDs, there re effetive heuristis for nmi reorering, se on loll swpping vriles. In this pper, we propose n nlogous pproh for SDDs whih nvigtes the spe of vtrees vi two opertions: one se on tree rottions n seon se on swpping hilren in vtree. We propose prtiulr heuristi for nmill serhing the spe of vtrees, showing tht it n fin SDDs tht re n orer-of-mgnitue more suint thn OBDDs foun nmi reorering. Introution A new representtion of Boolen funtions ws reentl propose, lle the Sententil Deision Digrm (SDD), whih generlizes the Orere Binr Deision Digrm (OBDD), n hs numer of interesting properties (Drwihe 2011; Xue, Choi, n Drwihe 2012). First, while eisions re performe on the stte of single vrile in OBDDs (i.e., literls), suh eisions re performe on the stte of multiple vriles in SDDs (i.e., sentenes). Seon, while n OBDD is hrterize totl vrile orer (Brnt 1986), n SDD is hrterize issetion of totl vrile orer, known s vtree. Despite this generlit, SDDs re still le to mintin numer of properties tht hve een ritil to the suess of OBDDs in prtie. For emple, SDDs re nonil (uner ertin onitions) n support n effiient ppl opertion whih llows one to omine SDDs using Boolen opertors. On the theoretil sie, n upper oun ws ientifie on the size of SDDs (se on treewith) (Drwihe 2011) tht is tighter thn the orresponing upper oun on the size of OBDDs (se on pthwith) (Prs, Chong, n Keutzer 1999; Hung n Drwihe 2004; Ferrr, Pn, n Vri 2005). Copright 2013, Assoition for the Avnement of Artifiil Intelligene ( All rights reserve. From prtil perspetive, OBDDs enefit gretl from vriet of heuristi lgorithms for fining vrile orers tht iel ompt OBDD representtions. For emple, sifting lgorithms, se on swpping neighoring vriles in totl vrile orer, hve een prtiulrl effetive t nvigting the spe of totl vrile orers (Ruell 1993). As n OBDD with prtiulr vrile orer orrespons to n SDD with prtiulr vtree, whih issets the orer, we n potentill fin even more ompt representtions if we evelop effetive heuristis for nvigting the spe of ll vtrees. In ft, (Xue, Choi, n Drwihe 2012) ientifie lss of Boolen funtions where ertin vrile orers le to eponentill lrge OBDDs, ut where prtiulr issetions of these orers le to SDDs of onl liner size. In this pper, we propose new gree serh lgorithm for optimizing vtrees. We introue two opertions tht re suffiient for nvigting the full spe of vtrees: one se on tree rottions, n nother se on swpping the hilren of vtree noe. Using the rottion n swpping opertions s primitives, we propose gree serh lgorithm tht n e use to fin goo vtrees, in mnner nlogous to methos tht swp neighoring vriles to serh for goo vrile orers. We evlute our nmi vtree serh lgorithm empirill, fining tht it n ientif SDDs tht re n orer of mgnitue more suint thn OBDDs foun the CUDD pkge (Somenzi 2004), using nmi vrile reorering. The nmi vtree serh lgorithm tht we evlute is further implemente in pulil ville SDD lirr. 1 Tehnil Preliminries Upper se letters (e.g., X) will e use to enote vriles n lower se letters to enote their instntitions (e.g., ). Bol upper se letters (e.g., X) will e use to enote sets of vriles n ol lower se letters to enote their instntitions (e.g., ). A Boolen funtion f over vriles Z, enote f(z), mps eh instntition z of vriles Z to 0 or 1. A trivil funtion mps ll its inputs to 0 (enote flse) or mps ll its inputs to 1 (enote true). 1 The SDD pkge is ville t s.ul.eu/s/

2 B A D () vtree 4 C 2 B A B! 6 C B 2 B A 5 D C D! () grphil epition of n SDD Figure 1: Funtion f = (A B) (B C) (C D). Consier Boolen funtion f(x, Y) with isjoint sets of vriles X n Y. If f(x, Y) = (p 1 (X) s 1 (Y))... (p n (X) s n (Y)) then the set {(p 1, s 1 ),..., (p n, s n )} is lle n (X, Y)- eomposition of the funtion f n eh pir (p i, s i ) is lle n element of the eomposition (Piptsriswt n Drwihe 2010). The eomposition is further lle n (X, Y)-prtition iff the p i s form prtition (Drwihe 2011). Tht is, p i flse for ll i; n p i p j = flse for i j; n i p i = true. In this se, eh p i is lle prime n eh s i is lle su. An (X, Y)-prtition is ompresse iff its sus re istint, i.e., s i s j for i j (Drwihe 2011). Compression n lws e ensure repetel isjoining the primes of equl sus. Moreover, funtion f(x, Y) hs unique, ompresse (X, Y)- prtition. Finll, the size of eomposition, or prtition, is the numer of its elements. Note tht (X, Y)-prtitions generlize Shnnon eompositions, whih fll s speil se when X ontins single vrile. OBDDs result from the reursive pplition of Shnnon eompositions, leing to eision noes tht rnh on the sttes of single vrile (i.e., literls). As we show net, SDDs result from the reursive pplition of (X, Y)-prtitions, leing to eision noes tht rnh on the stte of multiple vriles (i.e., ritrr sentenes). Sententil Deision Digrms (SDDs) Consier the full inr tree in Figure 1(), whih is known s vtree (v l n v r will e use to enote the left n right hilren of vtree noe). Consier lso the Boolen funtion f = (A B) (B C) (C D) over the sme vriles. Noe v = 6 is the vtree root. Its left sutree ontins vriles X = {A, B} n its right sutree ontins Y = {C, D}. Deomposing funtion f t noe v = 6 mounts to generting n (X, Y)-prtition of funtion f. The unique ompresse (X, Y)-prtition here is {(A }{{ B }, }{{} true), ( A }{{ B }, }{{} C ), ( }{{} B, } D {{ C } )} prime su prime su prime su This prtition is represente the root noe of Figure 1(). This noe, whih is irle, represents eision noe with three rnhes. Eh rnh orrespons to one element p s of the ove prtition. Here, the left o ontins 0 A 6 1 B C D () right-liner 6 A A 5 B B 4 C D C! () SDD B A 1 C D 0 () OBDD Figure 2: A vtree, SDD n OBDD for (A B) (C D). prime when the prime is literl or onstnt; otherwise, it ontins pointer to prime. Similrl, the right o ontins su or pointer to su. The three primes re eompose reursivel, ut using the vtree roote t v = 2. Similrl, the sus re eompose reursivel, using the vtree roote t v = 5. This reursive eomposition proess moves own one level in the vtree with eh reursion, terminting when it rehes lef vtree noes. The full SDD for this emple is epite in Figure 1(). A eision noe is si to e normlize for vtree noe v iff it represents n (X, Y)-prtition where X re the vriles of v l n Y re the vriles of v r. In Figure 1(), eh eision noe is lele with the vtree noe it is normlize for. The size of n SDD is the sum of sizes ttine its eision noes. The SDD in Figure 1() hs size 9. SDDs otine from the ove proess re lle ompresse iff the (X, Y)-prtition ompute t eh step is ompresse. These SDDs m ontin trivil eision noes whih orrespon to (X, Y)-prtitions of the form {(, α)} or {(α, ), ( α, )}. When these eision noes re remove ( ireting their prents to α), the resulting SDD is lle trimme. Compresse n trimme SDDs re nonil for given vtree (Drwihe 2011) 2 n we shll restrit our ttention to them in this pper. 3 SDDs support poltime ppl opertion, llowing one to omine two SDDs using n Boolen opertor (Drwihe 2011). 4 OBDDs orrespon to SDDs tht re onstrute using right-liner vtrees (Drwihe 2011). A right-liner vtree is one in whih eh left-hil is lef; see Figure 2(). When using suh vtrees, eh onstrute (X, Y)-prtition is suh tht X ontins single vrile, therefore, orresponing 2 Given the usul ssumption tht the SDD hs no isomorphi sugrphs, whih n e esil ensure in prtie using the unique-noe tehnique from the OBDD literture. 3 We will lso ssume reue OBDDs, whih re nonil for given vrile orer (Brnt 1986). 4 The poltime ppl oes not gurntee tht the resulting SDD is ompresse, et the ppl we utilize in this work ensures suh ompression s this hs prove ritil in prtie.

3 A A rr vnoe() w B D C D B C Figure 3: Two vtrees tht isset orer A, B, C, D. w () lr vnoe() () to Shnnon eomposition. In this se, primes re gurntee to lws e literls (i.e., vrile or its negtion). Moreover, eision noes re gurntee to e inr, leing to OBDDs (ut with ifferent snt); see Figure 2. Vtrees n Vrile Orers OBDDs re hrterize totl vrile orers, so serhing for ompt OBDD is one nvigting the spe of vrile orers. Similrl, SDDs re hrterize vtrees, so serhing for ompt SDD will e one nvigting the spe of vtrees. In ft, the spe of vtrees n e inue onsiering the issetions of ll vrile orers. Definition 1 (Dissetion) A vtree issets totl vrile orer π iff left-right trversl of the vtree visits leves (vriles) in the sme orer s π. Figure 3 epits two issetions of the sme vrile orer. The serh spe over vtrees n then e hrterize two imensions: totl vrile orers n their issetions. We hve n! totl vrile orers over n vriles. We lso hve C n 1 = (2n 2)! n!(n 1)! issetions of totl vrile orer over n vriles. 5 Hene, there re n! C n 1 = (2n 2)! (n 1)! totl vtrees over n vriles. The following tle gives sense of these serh spes in terms of the numer of vriles n. n # of orerings # of issetions # of vtrees It is well known tht the hoie of totl vrile orer n le to eponentil hnges in the size of orresponing OBDD n, hene, SDD. Moreover, it is known tht ifferent issetions of the sme vrile orer n le to eponentil hnges in the SDD size (Xue, Choi, n Drwihe 2012). In ft, this lst result is more speifi: rightliner issetion of ertin orers les to n SDD/OBDD of eponentil size, et some other issetion of the sme orer les to n SDD of onl liner size. This onl emphsizes the importne of serhing for goo vtrees. Nvigting the Spe of Vtrees When serhing for ompt OBDD, the spe of totl vrile orers is usull nvigte vi swps of neighoring vriles sine repete pplition of this opertion is gurntee to inue ll totl vrile orers (Knuth 2005). 5 C n 1 is the (n 1)-st Ctln numer, whih is the numer of full inr trees with n leves (Cmpell 1984). Figure 4: Rotting vtree noe right n left. Noes,, n m represent leves or sutrees. () swp vnoe() swp vnoe() Figure 6: Swpping the hilren of vtree noe, k n forth. Noes n m represent leves or sutrees. As we shll see net, two vtree opertions, lle rotte n swp, llow one to nvigte the spe of ll vtrees. Figure 4 illustrtes two rotte opertions on inr trees. The first is right rottion, enote rr vnoe(), n the seon is left rottion, enote lr vnoe(). These re inverse opertions tht nel eh other. Rottions re known to e suffiient for enumerting ll inr trees over n noes (Lus, vn Bronigien, n Ruske 1993). Moreover, rottions re known to keep the in-orering of noes in inr tree invrint. Hene, using rottions, one n enumerte ll issetions of given vrile orer. Figure 5 illustrtes how to enumerte ll issetions of vrile orers over 4 vriles, using rottions. Figure 6 illustrtes the swp opertion on inr trees, enote swp vnoe(). This opertion swithes the left n right hilren n. Upon performing swp opertion, seon swp will uno the first. Importntl, rottions n swp llow one to eplore ll vtrees. The proof rests on showing first how to swp two neighoring vriles, A n B, in the vrile orer of right-liner vtree. Suppose tht we hve suh vtree whih issets the orer π 1, A, B, π 2, where π 1 n π 2 re suorers (possil empt). We hve two ses. First se: A n B re hilren of the sme prent. In this se, π 2 must e empt n swp vnoe() will generte vtree tht issets the orer π 1, B, A. Seon se: A hs prent w n B hs prent. In this se, w must lso e prent of. Moreover, the opertions lr vnoe(), swp vnoe(w), n rr vnoe() will generte rightliner vtree tht issets the orer π 1, B, A, π 2. Suppose now tht we hve n ritrr vtree isseting n orer π 1 n we wish to nvigte to nother vtree tht issets ifferent orer π 2. Using rottions onl, we n nvigte to right-liner vtree tht issets orer π 1. B re- ()

4 () () () () (e) Figure 5: All 5 issetions of vrile orers over 4 vriles. Strting from vtree 5(), one otins vtrees 5(), 5(), 5(), 5(e), n then 5() gin vi the opertions lr vnoe(), lr vnoe(), lr vnoe(), rr vnoe() n rr vnoe(). pete pplition of the tehnique isusse ove, we n nvigte to nother right-liner vtree tht issets orer π 2. We n now nvigte to n other vtree tht issets orer π 2, using rottions onl. Hene, the rotte n swp opertions re omplete for nvigting the spe of ll vtrees. The SDD Pkge The rest of our isussion will nee to mke referene to our pulil ville implementtion, the SDD Pkge. The high-level interfe of this pkge n some of its rhiteture is highl influene the CUDD pkge for OBDDs. In prtiulr, it provies the following primitive opertions: ppl for onjoining or isjoining two SDDs, 6 negte for negting n SDD, 7 lr vnoe(), swp vnoe(), n rr vnoe() for performing the orresponing opertions on vtree noes n justing n orresponing SDDs oringl (more on this net). The pkge emplos onstruts tht re similr to those of the CUDD pkge, suh s mngers, unique-tles, omputtion hes, n grge olletor se on referene ounts. Our SDD pkge eposes ll these primitives together with soure oe for two lgorithms tht we isuss lter: vtree serh lgorithm, n CNF-to-SDD ompiler tht mkes nmi lls to our vtree serh lgorithm. First, however, we isuss the proess of justing n SDD fter the unerling vtree hs een hnge rottion or swpping. Ajusting SDDs uner Rotte n Swp Consier the SDD in Figure 1 n its orresponing vtree. Consier in prtiulr the eision noe normlize for vtree noe 6, whih orrespons to ompresse (AB, CD)- prtition. If we swp vtree noe 6, this eision noe must e juste so it orrespons to n equivlent n ompresse 6 The ppl opertion is se on the following result. If is Boolen opertor, n we hve two ompresse (X, Y)- prtitions {(p i, s i)} i n {(q j, r j)} j for funtions f n g, then {(p i q j, s i r j) p i q j flse} is n (X, Y)-prtition for funtion f g, lthough it m not e ompresse. Suessivel isjoining the primes of equl sus iels ompresse prtition. 7 The negte opertor is se on the following result. If {(p i, s i)} i is the ompresse (X, Y)-prtition for funtion f, then {(p i, s i)} i is the ompresse (X, Y)-prtition for f. (CD, AB)-prtition. 8 A similr justment is neee when rotting vtree noes. Consier for emple Figure 4 n suppose tht A, B n C re the vriles ppering in the vtrees roote t, n, respetivel. Upon right rottion, ll eision noes normlize for vtree noe in Figure 4() must e juste so the eome normlize for vtree noe w in Figure 4(). Tht is, these eision noes, whih orrespon to (AB, C)-prtitions, must e juste so the orrespon to equivlent (A, BC)-prtitions. Left rottion lls for similr justment, requiring one to onvert (A, BC)-prtitions into equivlent (AB, C)-prtitions. 9 Ajusting n SDD in response to vtree hnge is then mtter of onverting etween equivlent (X, Y)-prtitions. We will isuss these onversions net n show how the n e implemente using ppl n negte. The simplest onversion is from n (A, BC)-prtition to n (AB, C)-prtition (left rottion). Consier then ompresse (A, BC)-prtition {( 1, 1 ),..., ( n, n )}, where eh su i hs the ompresse (B, C)-prtition {( i1, i1 ),..., ( imi, imi )}. One n then show tht {( i ij, ij ) i = 1,..., n n j = 1,..., m i } is n equivlent (AB, C)-prtition, whih m not e ompresse. This prtition n e ompute using ppl to onjoin eisting SDD noes i n ij. Moreover, it n e ompresse isjoining the primes of equl sus, gin, using ppl. The net two onversions require one to ompute the Crtesin prout of formul-prtitions. In prtiulr, suppose tht {α 1,..., α n } is formul-prtition (i.e., α i α j = flse for i j, n α 1... α n = true). Suppose further tht {β 1,..., β m } is nother formul-prtition. The Crtesin prout is efine s {α i β j α i β j flse, i = 1,..., n, j = 1,..., m}. This prout is lso formulprtition n n e ompute using ppl. To see how to just n SDD for swp, onsier ompresse (X, Y)-prtition {(p 1, s 1 ),..., (p n, s n )}. We n onstrut the equivlent, ompresse (Y, X)-prtition s 8 This justment m le to the retion of new eision noes normlize for the esennts of vtree noe 6. It m lso le to removing referenes to eisting eisions noes. 9 Note tht eision noes normlize for vtree noe w in Figure 4() ontinue to e normlize for w fter right rottion. Similrl, eision noes normlize for vtree noe in Figure 4() ontinue to e normlize for fter left rottion.

5 follows. We first ompute the Crtesin prout of formulprtitions {s 1, s 1 },..., {s n, s n }, whih is gurntee to ontin the primes of our sought (Y, X)-prtition. Eh prime in this prout must orrespon to onjuntion of the form 1... n where eh i equls s i or s i. Let I e the inies i of ll i = s i. The orresponing su is then i I p i. One n show tht the esrie (Y, X)- eomposition is ompresse (Y, X)-prtition n equivlent to the originl (X, Y)-prtition. Moreover, it n e iretl ompute using ppl n negte. Finll, we onsier the justment of n SDD ue to right rottion. Consier ompresse (AB, C)-prtition {( 1, 1 ),..., ( n, n )}, where eh prime i hs the ompresse (A, B)-prtition {( i1, i1 ),..., ( imi, imi )}. We first ompute the Crtesin prout of formul-prtitions { 11,..., 1m1 },..., { n1,..., nmn }, whih is gurntee to ontin the primes of our sought (A, BC)-prtition. Eh prime in this prout orrespons to onjuntion of the form 1j1... njn. The orresponing su is then n i=1 ij i i. One n show tht the esrie (A, BC)- eomposition is n (A, BC)-prtition n equivlent to the originl (AB, C)-prtition, ut m not e ompresse. It n lso e ompute n ompresse using ppl. We lose this setion omprison to the proess of justing OBDDs fter swpping two neighoring vriles in totl vrile orer. Suh justments re known to hve oune impt on the OBDD size s it onl involves lol justment to the OBDD struture (Ruell 1993). Hene, when serhing for totl vrile orer using vrile swps, eh move in the serh spe is gurntee to e effiient. The sitution is ifferent for vtrees. In prtiulr, eh move in this spe (rottion or swp) involves nontrivil hnges to the SDD struture. In ft, (Xue, Choi, n Drwihe 2012) hs shown tht swpping the hilren of vtree noe n le to n eponentil hnge in the SDD size. Hene, while swpping two vriles in totl vrile orer n hve preitle, ut onservtive, effet on the OBDD size, swpping single pir of hilren in vtree n potentill otin signifintl more suint SDD in one opertion. From this perspetive, the potentill epensive swp opertion for vtrees llows one to mke lrge jumps in the serh spe, wheres the reltivel inepensive swp opertion in totl vrile orers m nee to e pplie mn times efore hieving the sme effet. Serhing for Goo Vtree Our serh lgorithm ssumes n eisting vtree n orresponing SDD. The lgorithm n e lle on n vtree noe v n it will tr to minimize the SDD size serhing for new su-vtree to reple the one urrentl roote t v. The lgorithm first mkes reursive lls on the hilren of vtree noe v. It then onsiers v n the two levels elow it (when pplile) s shown in Figure 7(). One n isolte two vtree frgments in this figure: the l-vtree in Figure 7() n the r-vtree in Figure 7(). The l-vtree is just one of 12 vtrees over leves, n. Similrl, the r-vtree is just one of 12 vtrees over leves, n. Eh one of these 24 vtrees les to vrition on the originl su-vtree v () vtree frgment v () l-vtree Figure 7: A vtree frgment. v () r-vtree roote t v. The propose serh lgorithm ttempts to nvigte through ll 24 vritions using pre-store sequene of rottions n swps whih is gurntee to le through them, returning k to the originl l-vtree or r-vtree tht we strt with. The lgorithm then hooses the one vrition with smllest SDD size 10 n nvigtes k to tht vrition. At this point, the lgorithm is si to hve omplete single pss on the su-vtree roote t v. If pss hnges the SDD size more thn given threshol (set to 1% in our eperiments), nother pss is me. Tht is, nother ll is me on vtree noe v with orresponing reursive lls on its hilren. Otherwise, the lgorithm termintes. We will now eplin the term ttempt use erlier. The SDD pkge llows the user to speif time or size limits for the rotte n swp opertions. If these limits re eeee while the opertion is tking ple, the opertion fils n the stte of the vtree n orresponing SDD re rolle k to how the eiste efore the opertion strte. In our eperiments, we use size limits ut not time limits. Thus, the lgorithm m not nvigte through ll 24 vritions esrie ove if the size limit is eeee. We use size limit of 25% for swp, using the opertion to fil if swpping vtree noe les to inresing the SDD size more thn 25%. The size limits for rottions re set to 75%. Dnmi Compiltion of CNFs into SDDs One tpill genertes n SDD inrementll n tries to minimize it if its size strts growing too muh uring the genertion proess. Consier for emple the proess of ompiling CNF into n SDD. One tpill strts with n initil vtree, ompiles eh luse of the CNF into orresponing SDD, n then onjoins these SDDs (using ppl). Sine these onjoin opertions tke ple in sequene, the finl SDD is then si to e onstrute inrementll. Tpill, if onjoin opertion grows the SDD size ertin ftor, one tries to serh for etter vtree efore proeeing with the remining onjoin opertions. This woul e the tpil usge of the serh lgorithm evelope in the previous setion. This woul lso e the proper ontet for evluting its effetiveness whih is lso the ontet usull use for evluting nmi orering heuristis for OBDDs. The eperiments of the net setion re thus onute in the ontet of ompiling CNFs to SDDs while using nmi vtree serh s isusse ove. 10 We rek ties preferring smller eision noe ounts n etter vtree lne.

6 Suppose we re given CNF s set of luses n vtree for the vriles of. Suppose further tht eh luse is ssigne to the lowest vtree noe v whih ontins the vriles of luse (noe v is unique). Our lgorithm for ompiling CNFs tkes this lele vtree s input. The initil vtree struture provies reursive prtitioning of the CNF luses, with eh noe v in the vtree hosting set of luses v. The lgorithm reursivel ompiles the luses hoste noes in the su-vtrees roote t the hilren of v. This les to two SDDs orresponing to these hilren. The lgorithm onjoins these two SDDs using ppl. It then itertes over the luses hoste t noe v, ompiling eh into n SDD, 11 n onjoining the result with the eisting SDD. If this lst onjoin opertion grows the SDD size more thn ertin ftor (sine the lst ll to vtree serh), the vtree serh lgorithm is lle gin on noe v. 12 In our eperiments, we set this ftor to 20%. We lso visit the luses hoste noe v oring to their length, with shorter luses visite first. Eperimentl Results We evlute our lgorithm on CNFs of omintionl iruits use in the CAD ommunit, n in prtiulr, from the LGSnth89, iss85 n iss89 enhmrk sets. We use the CUDD pkge to ompile CNFs to OB- DDs, using nmi vrile reorering n efult prmeters. 13 We further onjoin luses oring to the proeure from the previous setion, ssuming right-liner vtree inue the nturl vrile orer of the CNF. 14 For our SDD ompiler, we initill use lne vtree isseting the nturl vrile orer of the CNF. Eperiments on the LGSnth89 suite were performe on 2.67GHz Intel Xeon 5650 CPU with ess to 12GB RAM. Eperiments on the iss85 n iss89 suites were performe on 2.83GHz Intel Xeon 5440 CPU with ess to 8GB RAM. We elue enhmrks from these suites if (1) oth OBDD n SDD ompiltions file given two hour time limit, or (2) if the resulting SDD hs size of less thn 2,000, whih we onsier too trivil. Our eperimentl results in Tle 1 ll for numer of oservtions. First, the SDD turns out to e more ompt representtion in ll enhmrks suessfull ompile oth the SDD n OBDD ompilers. This is perhps 11 The SDD pkge provies primitive tht returns n SDD for given literl. Hene, one n esil ompile luse into n SDD isjoining the SDDs orresponing to its literls, using ppl. 12 In priniple, luse hoste t noe v, tht hs not et een onjoine, oul possil e re-ssigne to lower noe fter vtree serh is invoke. However, we woul hve lre visite these noes uring the ompiltion lgorithm, so we just finish onjoining those luses t noe v. 13 We use heuristi CUDD REORDER SYMM SIFT, whih is smmetri sifting (Pn, Somenzi, n Plessier 1994). We lso invoke sifting in post-proessing step, fter ompiltion. 14 In n erlier version of this pper, we ssume nturl orering of the luses, whih proue worse results for OBDD ompiltions. Moreover, these erlier evlutions lws pre-proesse the CNF using MINCE. Here, we onsier ompiltion with n without MINCE, leing to more reveling omprison. not too surprising s it hs een previousl oserve tht even rnom issetion of totl vrile orers ten to le to SDDs tht re more ompt thn the orresponing OBDDs (Drwihe 2011). We lso note tht in 11 of the enhmrks we evlute, we oserve t lest n orer-ofmgnitue improvement in size. This suggests tht the theoretil properties of SDDs isusse (Xue, Choi, n Drwihe 2012) n lso e relize in prtie. Moreover, in 4 instnes, SDD ompiltion sueee n OBDD ompiltion file. In 2 instnes, OBDD ompiltion sueee n SDD ompiltion file. Net, our seon oservtion is tht in mn enhmrks, our nmi ompiltion lgorithm ws fster thn CUDD s, n in 7 ses, orers-of-mgnitue fster. This is more evient in the more hllenging enhmrks with lrger ompiltions. This is prtiulrl interesting sine nmi vtree serh uses more epensive, et more powerful, opertions for nvigting its serh spe, in omprison to the effiient, ut less powerful, opertion use for nvigting totl vrile orers. We see similr results in Tle 2, where s preproessing step, we pplie MINCE to our CNFs, to fin n initil stti vrile orering for CUDD (Aloul, Mrkov, n Skllh 2004). For SDDs, we initill use lne vtree isseting the sme orer. We oserve, in generl, tht using MINCE orerings improves the resulting OBDD n SDD ompiltions, further llowing oth to suessfull ompile ses where the file to efore. We fin tht SDD ompiltions, in 4 ses, n still e n orer-of-mgnitue more suint. Moreover, there were 5 ses where SDD ompiltion sueee n OBDD ompiltion file. In totl, ross ll eperiments, there were 15 ses where the SDD ompiltion ws t lest n orer-of-mgnitue more suint thn the OBDD ompiltion. Moreover, there were 9 ses where SDD ompiltion sueee n OBDD ompiltion file, n there were 3 ses where OBDD ompiltion sueee n SDD ompiltion file. We finll sk: Is it the totl vrile orers isovere our vtree serh lgorithm, or the prtiulr issetion of these orers, whih is responsile for these fvorle results? To help nswer this question, we etrte the vrile orer emee in eh isovere vtree n then onstrute n OBDD using tht orer. The sizes of these OB- DDs re reporte in the olumn title r. liner SDD. In numer of ses, the resulting OBDDs re muh worse thn the OBDDs foun CUDD. These ses show emphtill tht issetion is wht eplins the improvements (i.e., mking eisions on ritrr sentenes inste of literls). Tht is, simpl isseting these OBDDs, we n otin even more suint SDDs, even if we isset n OBDD with poor vrile orer. In other ses, the resulting OBDD is omprle or more suint thn the OBDD foun CUDD. Interestingl, this suggests tht the vtree serh lgorithm tht we propose n lso fin effetive vrile orerings, in omprison to more speilize reorering heuristis.

7 size OBDD time OBDD r. liner OBDD CNF OBDD SDD SDD OBDD SDD SDD SDD SDD 9smml 54, C432 9, ,512 C432 out 228,036 5, , pe7 148,876 8, , ,440 8, , , ,456,186 22, , , , ht 12,296 4, , omp 2,774 2, , ount 55,312 3, , , emple2 34,456 8, , f51m 10,186 3, , frg1 358,216 81, , ll 31,082 6, , mu 3,052 2, , m er 19,632 2, , pm1 2,650 2, , st 9,954 8, , ttt2 244,598 14, , unreg 20,338 3, , v 43,626 4, z4ml 3,112 2, , s298 10,122 3, , s344 39,646 5, , s349 10,048 3, , s382 17,974 3, , s386 28,560 7, , s400 6,088 3, , s420 17,744 4, , s444 10,472 4, , s510 26,892 7, , s ,662 9, , s526n 107,334 6, , s ,688 10, , , s713 11, ,816 s ,652 23, , s ,934 9, , s ,470 14, , , ,092 9, , , ,400 1, ,193 2, Tle 1: OBDD n SDD ompiltions over LGSnth89, iss85 n iss89 suites. Missing entries inite file ompiltion, either n out-of-memor, or timeout of 2 hours. OBDD/SDD olumns report reltive improvement (in size or time). Bole tet inite ses where time/size improvements were n orer-of-mgnitue or more, or if SDD ompiltion sueee n OBDD ompiltion file. Reporte sizes re se on SDD nottion. Reporte times re in seons.

8 size OBDD time OBDD r. liner OBDD CNF OBDD SDD SDD OBDD SDD SDD SDD SDD 9smml 49,576 14, , C ,984 9, , C ,253 1, ,369,492 C ,549 1, C1908 3,216,974 5, lu2 38,298 11, , lu4 267,562 5, pe6 283,174 1, pe7 33,124 6, , ,270 10, , ,674 13, , ht 8,764 4, , ount 13,938 3, , emple2 15,044 7, , f51m 8,030 3, , frg1 374, , , frg2 255,136 6, ll 7,216 5, , mu 3,366 2, , m er 1,728 2, , st 7,154 6, , term1 1,484, , , ,930, ttt2 36,428 13, , unreg 39,590 3, , v 44,568 13, ,160 27, ,091, Tle 2: OBDD n SDD ompiltions over the LGSnth89 suite, using MINCE vrile orers. See lso Tle 1. Conlusion This pper provies the elements neessr for mking the sententil eision igrm (SDD) vile tool in prtie t lest in omprison to the influentil OBDD. In prtiulr, the pper shows how one m nmill serh for vtrees tht ttempt to minimize the size of onstrute SDDs. Our ontriutions inlue: (1) hrteriztion of the vtree serh spe s issetions of totl vrile orers; (2) n ientifition of two vtree opertions tht llow one to nvigte this serh spe; (3) orresponing opertions for justing n SDD ue to vtree hnges; (4) CNF to SDD ompiler se on nmi vtree serh; n (5) pulil ville SDD pkge tht emoies ll of the previous elements. Empirill, our results show tht our propose pproh for onstruting SDDs n le to orerof-mgnitue improvements in time n spe over similr pprohes for onstruting OBDDs. B relesing our SDD pkge to the puli, we hope tht the ommunit will hve the neessr infrstruture for uiling even more effetive serh lgorithms in the future. Aknowlegments This work hs een prtill supporte ONR grnt #N , NSF grnt #IIS , n NSF grnt #IIS Referenes Aloul, F. A.; Mrkov, I. L.; n Skllh, K. A Mine: A stti glol vrile-orering heuristi for SAT serh n BDD mnipultion. J. UCS 10(12): Brnt, R. E Grph-se lgorithms for Boolen funtion mnipultion. IEEE Trnstions on Computers C-35: Cmpell, D. M The omputtion of Ctln numers. Mthemtis Mgzine 57(4): Drwihe, A SDD: A new nonil representtion of propositionl knowlege ses. In IJCAI, Ferrr, A.; Pn, G.; n Vri, M. Y Treewith in verifition: Lol vs. glol. In LPAR, Hung, J., n Drwihe, A Using DPLL for effiient OBDD onstrution. In SAT. Knuth, D. E The Art of Computer Progrmming, Volume 4, Fsile 2: Generting All Tuples n Permuttions. Aison- Wesle Professionl. Lus, J. M.; vn Bronigien, D. R.; n Ruske, F On rottions n the genertion of inr trees. J. Algorithms 15(3): Pn, S.; Somenzi, F.; n Plessier, B Smmetr etetion n nmi vrile orering of eision igrms. In ICCAD, Piptsriswt, K., n Drwihe, A A lower oun on the size of eomposle negtion norml form. In AAAI. Prs, M. R.; Chong, P.; n Keutzer, K Wh is ATPG es? In DAC, Ruell, R Dnmi vrile orering for Orere Binr Deision Digrms. In ICCAD, Somenzi, F CUDD: CU eision igrm pkge. http: //vlsi.oloro.eu/ fio/cudd/. Xue, Y.; Choi, A.; n Drwihe, A Bsing eisions on sentenes in eision igrms. In AAAI,