Backtrack Search Using ZBDDs

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1 Bktrk erh Using ZBDDs Fi A. Aloul, Mher N. Mneimneh, Krem A. kllh {floul, mherm, Eletril Engineering n Computer iene University of Mihign Astrt We introue new pproh to stisfiility tht omines ktrk serh tehniques n zero-suppresse inry eision igrms (ZBDDs. his pproh impliitly represents stisfiility prolems using ZBDDs, n performs serh using opertions on this representtion. his methoology whih pts ktrk serh lgorithms to suh impliit representtions shoul llow for potentil eponentil inrese in the size of prolems tht n e hnle. We esrie how to perform ktrk serh n onflit ignosis with ZBDDs use s n unerlying struture for luse representtion. We lso report on our initil eperiments with this pproh. Introution Boolen tisfiility (A serves s n unerlying moel for wie rnge of pplitions in Computer iene, Artifiil Intelligene n Eletril Engineering, to nme few. Over the yers, this prolem hs een etensively investigte n effiient lgorithmi solutions egerly sought. A reserh efforts ulminte in n etensive olletion of propose solutions. Of these, the most known omplete lgorithms re se on the Dvis-utnm metho [2], n vritions of the Dvis-Logemnn-Loveln metho [3]. Despite literture onfusion, the two pprohes re ifferent. he former, se on resolution, performs eistentil elimintion on the propositionl vriles. he proeure is repete until the formul equls either (unstisfile prolem instne or (stisfile prolem instne. Resolution tens to e memory intensive s eistentil elimintion often genertes lrge numer of luses. he ltter pproh, se on ktrk serh, impliitly enumertes the spe of possile inry ssignments looking for stisfying one. A eision tree keeps trk of urrent ssignments n prunes the serh y itertively pplying unit propgtion, usully referre to s Boolen Constrint ropgtion [2]. If onflit is rehe, the serh ktrks to some previous ssignment. Conflit nlysis [7], n reursive lerning [5] omprise mjor enhnements to the si ktrk serh proeure. Conflit nlysis omes into ply when onflit rises, n s equte informtion, onflit luse, tht ntiiptes the possile reourrene of this onflit. Furthermore, onflit nlysis llows the serh proess to ktrk non-hronologilly to erlier levels in the serh tree, onsierly pruning the serh spe. On the other hn, reursive lerning, when etene to onjuntive norml form (CNF luses, ientifies neessry ssignments y emining the ifferent possile wys of stisfying given luse from the set of unssigne literls. hese improvements llowe solving lrge prolem instnes in vrious omins [7]. However, serh-se pprohes re still inple of hnling very lrge prolems rising from vrious EDA pplitions [] s they ten to epliitly represent the luse tse. his epliit representtion n enumertion often results in time n memory eplosion. Reently, this prolem hs een resse y Chtli et l. [] who propose implementing resolution using zerosuppresse inry eision igrms (ZBDDs [6, 9] s the unerlying t struture for luse enoing. heir pproh ws ple of solving two hr prolems tht known A solvers file t. he high ompression power of the unerlying t struture resulte in enormous reutions in lgorithmi ompleity. In this pper, we push the ove pproh further. We eplore using suh n impliit luse tse representtion with ktrk serh tehniques. his is motivte y the esire to integrte the vntges of BDD-se n Ase pprohes in hyri sheme. In ition, we show how to effiiently ientify onflits in suh t struture n how to generte onflit luses using resolution. 2 reliminries ZBDDs [6, 9] were inspire y the nee to effiiently represent n mnipulte sets of omintions. It is irete yli grph (DAG onsisting of two terminl noes, the - terminl (the empty set n the -terminl (the set of single empty omintion, n non-terminl noes eh of whih hs two hilren, the -suessor n the -suessor. In ition, eh non-terminl noe is lele with Boolen vrile. Given universe U {, 2,, n } of n ojets, omintion C (, 2,, m of m ojets from U n e represente y n n -it inry vetor X (, 2,, n where if ojet i is in C, n otherwise, i n. A set of omintions n e represente y hrteristi funtion χ : {,} n {, } where χ ( X if X n otherwise, X {, } n. In wht follows, we use set n its hrteristi funtion χ interhngely. ZBDD noe semntis re illustrte in Figure (. If noe v with lel represents set, n v s -suessor n -suessor represent n respetively, then ( { i },where: A B { ( A n ( B }. (

2 ( { } ( { } ( / j k j k j j / ( ( Figure. ( ZBDD noe semntis ( noe merging rule n ( noe elimintion rule ZBDD onstrution is se on two reution rules illustrte in Figure ( n (. he noe merging rule merges two noes if they hve the sme lel n ientil - n - suessors, wheres the noe elimintion rule elimintes noe if its -suessor is the -terminl. Eh pth from the root noe to the -terminl orrespons to one omintion C of where if no noe lele eists long tht pth. It is this property tht reners ZBDDs ompt representtion for sprse omintions. As n emple onsier the universe U {,,,, e, f, g} n the set: {(,,,( g,,,(,,,( e,, g (2 ( e,,,( e,, h, ( fg,,, ( fgh,, } n e represente y the ZBDD shown in Figure 3(. (Note tht we lel the ZBDD noes with the ojets nmes inste of their enoings for ese of presenttion. Minto [9] presente effiient lgorithms tht implement set theoreti opertions on ZBDDs. hese opertions inlue union, intersetion, ifferene, nprout, mong others. With effiient hing tehniques, these lgorithms n eeute in time proportionl to the ZBDD size rther thn the rinlity of the omintion sets. It ws emonstrte in [] tht the ove pproh n e etene to effiiently enoe sets of luses. In this se, eh vrile n its omplement re ojets of U, n eh pth from the root to the -terminl orrespons to single luse. he numer of pths to the -terminl equls the numer of luses in the luse tse. As n emple, the set of luses: ( ( ( l ( ( ( (3 ( ( orrespons to set of omintions from U l {,,,,,,, } n n e represente y the ZBDD shown in Figure 3(. Using this pproh, the semntis of Boolen Alger, suh s susumption, n e superimpose on ZBDD reution rules to hieve further ompression. As n emple onsier the ZBDD illustrte in Figure 2( where the -suessor n the -suessor of the root re ientil. Using ZBDD noe semntis, ( n y the susumption rule of Boolen Alger,. Another CNF-speifi reution rule is susume ifferene []: given two sets n, the susume ifferene of y, enote s, is the set of ( ( ( Figure. 2 ( luse susumption rule n ( susumption elimintion rule luses of tht re not susume y ny luse from. In ZBDD, whose root noe represents the set n its n -suessors represent n respetively,. ine is inepenent of, luses in n susume luses in, while luses in n t susume ny luse in. Consequently, [ ]. his reution rule is illustrte in Figure 2(. It ws shown tht the reursive pplition of this rule results in ZBDD tht is free of susume luses. In ition, susume ifferene n e use s the uiling lok for susumption-free union n susumption-free prout opertions []. hese opertions were inorporte in multi-resolution version of the D proeure for stisfiility. Given luse tse ϕ n the hrteristi funtion χ ϕ enoing the luses of ϕ, fter vrile s selete for eistentil elimintion, multi-resolution uses stnr ZBDD opertions to prtition the luse set into three sets: ϕ, ϕ n ϕ. ϕ enotes the set of luses hving the literl, ϕ the set of luses hving the literl, n ϕ the set of luses hving neither nor. Eistentil elimintion is performe s follows:.ϕ ( ϕ ϕ where ontes the oftor ϕ ϕ of ϕ with respet to. he union etween ϕ n ϕ trnsltes into susumption-free prout on the ZBDDs representing their hrteristi funtions n the intersetion of the result with ϕ orrespons to susumption-free union on the orresponing ZBDDs. he vntge of multi-resolution is the reution in lgorithmi ompleity of the opertions. Eh of the ove opertions epen on the sizes of the orresponing ZBDDs, mesure in numer of noes, n not the size of the luse set (i.e literls they enoe. 3 ZBDDs As truture for Bktrk erh Despite vrious lgorithmi solutions, ktrk serh remins the most prevlent tehnique for ttking the stisfiility prolem. Bktrk serh impliitly enumertes the spe of truth ssignments using eision tree to mintin urrent ssignments to Boolen vriles. Although mny effetive improvements were inorporte in the lgorithm, it is still inompetent for lrge-sle luse tses euse of its epliit representtion of luses. o onquer this, we propose n impliit ktrk serh lgorithm tht uses ZBDDs s the unerlying t struture for luse rep- 2

3 e f g g h ( ( Figure. 3 ( ZBDD representing ( ZBDD representing l ( ZBDD representing n ( ZBDD representing l fter pplying reution rules ( new l ( resenttion. A generi ktrk serh lgorithm is omprise of three min engines [8]: Deie(, Deue(,nDignose(. In wht follows, we riefly esrie eh engine n illustrte how to hieve its funtion when ZBDDs re the unerlying t struture. Deie( uses heuristi knowlege to mke eletive ssignments to vriles. When using ZBDDs, we ssign vrile y ing one-literl luse, representing this ssignment to the luse set. his is hieve y unioning, using susumption-free union, the hrteristi funtion of the luse to e e with the hrteristi funtion of the luse tse. As n emple, onsier l gin. o ssign to, we the luse ( to l to get: ( ( ( l (4 ( ( ( the three luses (, (, n ( re susume y (. Deue( etermines the onsequenes of the ssignments elete y Deie(, typilly yieling itionl fore ssignments to, i.e. implitions of, other vriles. his is hieve y repetely pplying the unit luse rule until no unit luses eist. o implement this pproh using ZBDDs, we nee to ientify unit luses n sor on them. ingle-literl luses re ientifie y reursively trversing zero-suessors of the ZBDD enoing the luse set strting t the root; ny noe whose -suessor is the -terminl enotes unit luse. Asorption is then rrie on the ientifie set of unit luses y reursively pplying the sorption reution rule illustrte in Figure 4. he reursive pplition of this proeure utomtilly hnles unit propgtion, eliminting the nee to trk implitions epliitly. An empty luse, i.e. -terminl, esigntes onflit, initing the eistene of n unstisfie luse, while set onsisting of only unit luses represents stisfying ssignment. Applying sorption to, we get: l 2 l Asor on Figure. 4 ZBDD sorption rule ( ( ( ( ( ( Dignose( hnles the ourrene of onflits n ktrks ppropritely to previous eision. Besies, it nlyzes the uses of the onflit n genertes equte informtion to prevent its re-ourrene. While our Deue( engine elimintes the nee for n implition grph tht keeps trk of ssignments, it lks the ility to ientify vrile ssignments using onflits. o surmount this, we keep opy of the luse tse efore eh itertion of the unit propgtion rule. On onflit, we use this opy to ientify onfliting vriles. his is hieve y heking for pttern of the following struture 2 k (.heeisteneof 2 k suh pttern inites tht eh of to k is onfliting vrile n n now e use to generte the lerne luses, tht n effetively prune the serh spe. We use novel onflit ignosis pproh tht impliitly genertes lerne luses y pplying resolution on ientifie onfliting vriles. As n emple, onsier gin. Assume tht the urrent ssignments re, n the Deie( engine selets. his results in onf ( ( initing tht is onflit vrile. Applying resolution on results in two luses: ( n (. hese luses re eto l toget: ( ( ( new l (6 ( ( ( (5 3

4 ( ( ( ( Instne ime Regulr M- CL ime Reue M- CL Comp ression im-5-3_4-yes-2 > im-5-_6-no im-2-_6-yes- > jnh2 > 2374 > pret6_ uois hole ii pr pret6_ ss hole E8 hole 2.7 E 2.5 E 3.39E4 Figure. 5 Boolen lgeri mnipultion tehniques whose ZBDD is shown in Figure 3(. he lgorithm ontinues y unioning previous eision i.e, ( ( with new l. ine knowing more is goo, ugmenting the formul with itionl luses n proue n vlnhe of implitions uring serh, onsequently leing to rmti reution in serh time. However, knowing too muh is sine pplying lin resolution n generte n eponentil numer of luses tremenously slowing own the serh performne. herefore, we impose limit on the size (numer of literls of the generte luses s heuristi for limiting the numer of generte luses. Also, we seletively perform resolution on susets of the luse tse tht only onsists of previously ssigne vriles. he set of generte luses re e to the initil luse tse using susumption-free union. Further reution tehniques, illustrte in Figure 5, tht perform omintion of Boolen lgeri mnipultions (i.e. sorption, susumption, n resolution n e lolly pplie to the ZBDD in orer to reue its size n eventully the numer of literls in the prolem. Figure 3( shows the result of suh reltions on luse set l : wheres the originl prolem h 8 luses, 24 literls, n require 3 ZBDD noes, the reue formul onsiste of 5 luses, literls, n require only 9 ZBDD noes. he time ompleity of the esrie lgorithms is funtion of the numer of noes in the ZBDD rther thn the size of the luse tse. With the strong ompression power of ZBDDs, this pproh promises etter results thn epliit tehniques when eling with lrge sle prolems. 4 Eperimentl Results o hek the effiieny of resolution using ZBDDs, we implemente version of ZRE []. Our lgorithm is implemente in C++ n uses the CUDD pkge [] to uil the le : Resolution Results ZBDDs. le shows the results for selete prolems from the DIMAC set [4]. All eperiments were onute on entium-ii 333 MHz mhine running Linu n equippe with 52 MB of RAM. he time-out limit ws set to seons. In generl, the resolution pproh ws ineffiient in solving the mjority of smll enhmrks n unle to solve ny of the lrge enhmrks. With itionl reution tehniques, we were le to improve the runtimes ut mny prolems remine unsolve. le lso shows the ompression pility, shown in the lst olumn, otine when using ZBDDs s oppose to epliit lists of luses. he ompression power is mesure s the rtio of the numer of literl in the prolem n the numer of noes in the ZBDD. Clerly, this impliit representtion provies gret memory reution, espeilly on struture prolems. It is instrutive to point out thtthe hole-n results reporte in Chtli et l. [], shown t the en of le, hs prtiulr struture tht n e represente very effiiently using speifi ZBDD vrile orer. In ontrst, the results shown in le were generte using fie ZBDD vrile orer ( 2,,, k. In generl, struture prolems suggests nturl effiient vrile orering for ZBDD onstrution s well s resolution vrile elimintion. Figure 6 shows the serh runtimes for vrious size limits of the luses generte uring onflit nlysis for four enhmrks. he grphs lso show the runtimes for the regulr n seletive pproh (isusse in etion 3 whih pplies resolution on the omplete prolem n on selete suset of the prolem, respetively. he seletive/reue urve represents omintion of the seletive pproh with the reution tehniques shown in Figure 5. he t lerly shows the vntge of limiting the numer of the generte luses n their sizes on the performne of the serh proess. An optiml limit eists for eh of the presente enhmrks, whih woul yiel miniml serh time. 4

5 im-5-3_4-yes-2.nf uois22.nf Regulr eletive eletive/reue Regulr eletive eletive/reue Runtime (se Runtime (se Mimum luse size ( Mimum luse size ( jnh2.nf im-2-_6-yes-.nf Regulr eletive eletive/reue Regulr eletive eletive/reue Runtime (se Runtime (se Mimum luse size ( Mimum luse size ( Figure. 6 Bktrk serh runtimes using ZBDDs However, unrestrite luse size limits n le to n eponentil numer of luses, hene severely slowing the serh performne. Although the shown enhmrks were esily solve y other A solvers, we onjeture tht further refinements of the serh lgorithm, espeilly onflit ignosis, shoul e le to hnle very lrge prolems tht re out of the sope of urrent solvers. uh refinements inlue, effiiently orering ZBDD vriles n ynmilly limiting luse size. 5 Conlusion We hve propose new pproh to stisfiility tht omines ktrk serh n resolution using ZBDDs. he vntge of suh n pproh is twofol. Firstly, it promises to e le to el with lrge sle luse tses through its impliit representtion. eonly, it serves s mens to stuy the time-spe treoff etween ktrk serh n resolution. Our preliminry results show the effetive memory ompression hieve with this pproh. Further work involves enhning this tehnique to hnle lrge prolems in fesile time. 6 Referenes []. Chtli n L. imon, Multi-Resolution on Compresse ets of Cluses, in ro. of the Int l Conferene on ools with Artifiil Intelligene, 2. [2] M. Dvis n H. utnm, A Computing roeure untifition heory, injournloftheacm, vol. 7, pp. 2-25, 96. [3] M. Dvis G. Logemnn, D. Loveln, A Mhine rogrm for heorem roving, in Communitions of the ACM, 962. [4] DIMAC Chllenge enhmrks in ftp://dims.rutgers.edu/pu/hllenge/st/enhmrks/nf. [5] W. Kunz n D. toffel, Resoning in Boolen Networks, Kluwer Aemi ulishers, Boston, MA, 997. [6] M. Loing, O. hroer, n I. Wegner, he heory of Zero- uppresse BDDs n the Numer of Knight's ours, in IFI WG.5 Workshop on Applitions of the Ree-Muller Epnsion in Ciruit Design, 995. [7] J.. Mrques-ilv n K. kllh, Boolen tisfiility in Eletroni Design Automtion, in ro. of the Design Automtion Conferene, 2. [8] J.. Mrques-ilv n K. kllh, GRA: A erh Algorithm for ropositionl tisfiility, in IEEE rnstions on Computers, vol. 48, no. 5, pp , 999. [9]. Minto, Zero-uppresse BDDs for et Mnipultion in Comintoril rolems, in ro. of the Design Automtion Conferene, 993. [] G. Nm, K. kllh, n R. Rutenr, tisfiility-bse Lyout Revisite: Detile Routing of Comple FGAs Vi erh-bse Boolen A, in the ro. of the Int l ymposium on Fiel rogrmmle Gte Arrys, 999. [] F. omenzi, CUDD: CU Deision Digrm kge, University of Coloro t Bouler, ftp://vlsi.oloro.eu/pu/. [2] R. Zih n D. A. MAllester, A Rerrngement erh trtegy for Determining ropositionl tisfiility, in ro. of the Ntionl Conf. on Artifiil Intelligene,