March eq Implementing Additional Reasoning into an Efficient Look-Ahead SAT Solver

Transcription

1 Mrh eq Implementing Additionl Resoning into n Effiient Look-Ahed SAT Solver Mrijn Heule, Mrk Dufour, Joris vn Zwieten nd Hns vn Mren Deprtment of Informtion Systems nd Algorithms, Fulty of Eletril Engineering, Mthemtis nd Computer Sienes, Delft University of Tehnology mrijn@heule.nl, m.dufour@student.tudelft.nl, zwieten@h.tudelft.nl, h.vnmren@its.tudelft.nl Astrt. This pper disusses severl tehniques to mke the lookhed rhiteture for stisfiility (St) solvers more ompetitive. Our ontriution onsists of redution of the omputtionl osts to perform look-hed nd hep integrtion of oth equivlene resoning nd lol lerning. Most proposed tehniques re illustrted with experimentl results of their implementtion in our solver mrh eq. 1 Introdution Look-hed St solvers usully onsist of simple DPLL lgorithm [5] nd more sophistited look-hed proedure to determine n effetive rnh vrile. The look-hed proedure mesures the effetiveness of vriles y performing look-hed on set of vriles nd evluting the redution of the formul. We refer to the look-hed on literl x s the Itertive Unit Propgtion (IUP) on the union of formul with the unit luse x (in short IUP(F {x})). The effetiveness of vrile x i is otined using look-hed evlution funtion (in short Diff), whih evlutes the differenes etween F nd the redued formul fter IUP(F {x i }) nd IUP(F { x i }). A widely used Diff ounts the newly reted inry luses. Besides the seletion of rnh vrile, the look-hed proedure my detet filed literls: if the look-hed on x results in onflit, x is fored to true. Detetion of filed literls n result in sustntil redution of the DPLL-tree. During the lst dede, severl enhnements hve een proposed to mke look-hed St solvers more powerful. In stz y Li [9] pre-seletion heuristis prop z re used, whih restrit the numer of vriles tht enter the look-hed Supported y the Duth Orgniztion for Sientifi Reserh (NWO) under grnt H.H. Hoos nd D.G. Mithell (Eds.): SAT 2004, LNCS 3542, pp , Springer-Verlg Berlin Heidelerg 2005

2 346 M. Heule et l. proedure. Espeilly on rndom instnes the pplition of these heuristis results in ler performne gin. However, the use of these heuristis is not ler from generl viewpoint. Experiments with our pre-seletion heuristis show tht different enhmrk fmilies require different numers of vriles entering the look-hed phse to perform optimlly. Sine muh resoning is lredy performed t eh node of the DPLL-tree, it is reltively hep to extend the look-hed with (some) dditionl resoning. For instne: integrtion of equivlene resoning in stz - implemented in eqstz [10] - mde it possile to solve vrious rfted nd rel-world prolems whih were eyond the reh of existing tehniques. However, the performne my drop signifintly on some prolems, due to the integrted equivlene resoning. Our vrint of equivlene resoning extends the set of prolems whih enefit from its integrtion nd ims to remove the disdvntges. Another form of dditionl resoning is implemented in the OKsolver 1 [8]: lol lerning. When performing look-hed on x, ny unit luse y i tht is found mens tht the inry luse x y i is implied y the formul, nd n e lerned, i.e. dded to the urrent formul. As with equivlene resoning, ddition of these lol lerned resolvents ould oth inrese nd derese the performne (depending on the formul). We propose prtil ddition of these resolvents whih results in speed-up prtilly everywhere. Generlly, look-hed St solvers re effetive on reltively smll, hrd formuls. Le Berre proposes [2] wide rnge of enhnements of the look-hed proedure. Most of them re implemented in mrh eq. Due to the high omputtionl osts of the n enhned look-hed proedure, elorte prolems re often solved more effiiently y other tehniques. Reduing these osts is essentil for mking look-hed tehniques more ompetitive on wider rnge of enhmrks prolems. In this pper, we suggest (1) severl tehniques to redue these osts nd (2) hep integrtion of dditionl resoning. Due to the ltter, enhmrks tht do not profit from dditionl resoning will not tke signifintly more time to solve. Most topis disussed in this pper re illustrted with experimentl results showing the performne gins y our proposed tehniques. The enhmrks rnge from uniform rndom 3-St ner the threshold [1], to ounded model heking (longmult [4], zrps [3]), ftoring prolems (pyhl run [12]) nd rfted prolems (stnion/hw [3], qusigroup [14]). Only unstisfile instnes were seleted to provide more stle overview. Comprison of the performne of mrh eq with performnes of stte-of-the-rt solvers is presented in [7], whih ppers in the sme volume. All tehniques hve een implemented into referene vrint of mrh eq, whih is essentilly slightly optimised version of mrh eq 100, the solver tht won two tegories of the St 2004 ompetition [11]. This vrint uses extly the sme tehniques s the winning vrint: full (100%) look-hed, ddition of ll onstrint resolvents, tree-sed look-hed, equivlene resoning, nd removl of intive luses. All these tehniques re disussed elow. 1 Version 1.2 t soliver/oksolver.html

3 Mrh eq Trnsltion to 3-St The trnsltion of the input formul to 3-St stems from n erly version of mrh eq, in whih it ws essentil to llow fst omputtion of the pre-seletion heuristis. Trnsltion is not required for the urrent pre-seletion heuristis, yet it is still used, euse it enles signifint optimistion of the internl dt strutures. The formul is pre-proessed to redue the mount of redundny introdued y strightforwrd 3-St trnsltion. Eh pir of literls tht ours more thn one together in luse in the formul is sustituted y single dummy vrile, strting with the most frequently ourring pir. Three luses re dded for eh dummy vrile to mke it logilly equivlent to the disjuntion of the pir of literls it sustitutes. In the following exmple x 2 x 4 is the most ourring literl pir nd is therefore repled with the dummy vrile d 1. x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 6 x 1 x 2 x 3 x 4 x 6 x 1 x 2 x 4 x 5 x 6 x 1 d 1 x 3 x 5 x 1 d 1 x 3 x 6 x 1 d 1 x 3 x 6 x 1 d 1 x 5 x 6 d 1 x 2 d 1 x 4 d 1 x 2 x 4 It ppers tht to hieve good performne, inry luses otined from the originl ternry luses should e given more weight thn inry luses otined from ternry luses whih were generted y trnsltion. This is omplished y n pproprite look-hed evlution funtion, suh s the vrint of Diff proposed y Duois et l. [6], whih weighs ll newly reted inry luses. 3 Time Stmps Mrh eq uses time stmp dt struture, TimeAssignments (TA), whih redues ktrking during the look-hed phse to single integer ddition: inresing the CurrentTimeStmp (CTS). All the vriles tht re ssigned during look-hed on literl x re stmped: if vrile is ssigned the vlue true, it is stmped with the CTS; if it is ssigned the vlue flse, it is stmped with CTS + 1. Therefore, simply dding 2 to the CTS unssigns ll ssigned vriles. The tul truth vlue tht is ssigned to vrile is not stored in the dt struture, ut n e derived from the time stmp of the vrile: stmp < CTS unfixed TA[x] = stmp CTS nd stmp 0 (mod 2) true stmp CTS nd stmp 1 (mod 2) flse Vriles tht hve lredy een ssigned efore the strt of the look-hed phse, i.e. during the solving phse, hve een stmped with the Mximum- TimeStmp (MTS) or with MTS + 1. These vriles n e unssigned y

4 348 M. Heule et l. stmping them with the vlue zero, whih hppens while ktrking during the solving phse (i.e. not during the look-hed phse). The MTS equls the mximl even vlue of n (32-it) integer. One hs to ensure tht the CTS is lwys smller thn the MTS. This will usully e the se nd it n esily e heked t the strt of eh look-hed. 4 Constrint Resolvents As mentioned in the introdution, inry resolvent ould e dded for every unry luse tht is reted during the propgtion of look-hed literl - provided tht the inry luse does not lredy exist. A speil type of resolvent is reted from unry luse tht ws ternry luse prior to the look-hed. In this se we spek of onstrint resolvents. Constrint resolvents hve the property tht they nnot e found y lookhed on the omplement of the unry luse. Adding these onstrint resolvents results in more vigorous detetion of filed literls. An exmple: First, onsider only the originl luses of n exmple formul (figure 1 ()). A look-hed on r, IUP(F { r}), results in the unry luse x. Therefore, one ould dd the resolvent r x to the formul. Sine the unry luse x ws originlly ternry luse (efore the look-hed on r), this is onstrint resolvent. The unique property of onstrints resolvents is tht when they re dded to the formul, look-hed on the omplement of the unry luse results in the omplement of the look-hed-literl. Without this ddition this would not e the se. Applying this to the exmple: fter ddition of r x to the formul, IUP(F { x}) will result in unry luse r, while without this ddition it will not. IUP(F { r}) lso results in unry luse t. Therefore, resolvent r t ould e dded to the formul. Sine unry luse t ws originlly inry luse, r t is not onstrint resolvent. IUP(F {t}) would result in unry luse r. r s s t s t x u v u w v w x r u x } } r x u x x () () () Fig. 1. Detetion of filed literl y dding onstrint resolvents. () The originl luses, () onstrint resolvents nd () fored literl

5 Mrh eq 349 Tle 1. Performne of mrh eq on severl enhmrks with three different settings of ddition of resolvents during the look-hed phse no resolvents ll inry ll onstrint resolvents resolvents Benhmrks time(s) treesize time(s) treesize time(s) treesize rndom unst 250 (100) rndom unst 350 (100) stnion/hw-n stnion/hw-n stnion/hw-n longmult longmult longmult pyhl-unst pyhl-unst qusigroup qusigroup qusigroup zrps/rule dt > zrps/rule dt > > Constrint resolvent u x is deteted during IUP(F { u}). After the ddition of oth onstrint resolvents (figure 1 ()), the look-hed IUP(F { x}) results in onflit, mking x filed literl nd thus fores x. Oviously, IUP(F { x}) will not result in onflit if the onstrint resolvents r x nd u x were not dded oth. Tle 1 shows the usefulness of the onept of onstrint resolvents: in ll our experiments, the ddition of mere onstrint resolvents outperformed vrint with full lol lerning (dding ll inry resolvents). This ould e explined y the ove exmple: dding other resolvents thn onstrint resolvents will not inrese the numer of deteted filed literls. These resolvents merely inrese the omputtionl osts. This explntion is supported y the dt in the tle: the tree-size of oth vrints is omprle. When we look t zrps/rule dt, it ppers tht only dding onstrint resolvents is essentil to solve this enhmrk within 2000 seonds. The node-ount of zero mens tht the instne is found unstisfile during the first exeution of the look-hed proedure. 5 Implition Arrys Due to the 3-St trnsltion the dt struture of mrh eq only needs to ommodte inry nd ternry luses. We will use the following formul s n exmple: F exmple = ( ) ( d) ( d) ( d) ( d) ( )

6 350 M. Heule et l. d d d d d d d d d d (i) Fig. 2. The inry (i) nd ternry (ii) implition rrys tht represent the exmple formul F exmple (ii) # luse 0 1 d 2 d 3 d 4 d 5 d d (i) Fig. 3. A ommon luse dtse / vrile index dt struture. All luses re stored in luse dtse (i), nd for eh literl the vrile index lists the luses in whih it ours (ii) Binry nd ternry luses re stored seprtely in two implition rrys. A inry luse is stored s two implitions: is stored in the inry implition rry of nd is stored in the inry implition rry of. A ternry luse ( d) is stored s three implitions: d is stored in the ternry implition rry of nd the similr is done for nd d. Figure 2 shows the implition rrys tht represent the exmple formul F exmple. Storing inry luses in implition rrys requires only hlf the memory tht would e needed to store them in n ordinry luse dtse / vrile index dt struture. (See figure 3.) Sine mrh eq dds mny inry resolvents during the solving phse, the inry luses on verge outnumer the ternry luses. Therefore, storing these inry luses in implition rrys signifintly redues the totl mount of memory used y mrh eq. Furthermore, the implition rrys improve dt lolity. This often leds to speed-up due to etter usge of the he. Mrh eq uses vrint of itertive unit propgtion (IFIUP) tht propgtes inry implitions efore ternry implitions. The first step of this proedure is to ssign s mny vriles s possile using only the inry implition rrys. Then, if no onflit is found, the ternry implition rry of eh vrile tht ws ssigned in the first step is evluted. We will illustrte this seond step with n exmple. (ii)

7 Mrh eq 351 Suppose look-hed is performed on. The ternry implition rry of ontins ( ). Now there re five possiilities: 1. If the luse is lredy stisfied, i.e. hs lredy een ssigned the vlue flse or hs lredy een ssigned the vlue true, then nothing needs to e done. 2. If hs lredy een ssigned the vlue true, then is implied nd so is ssigned the vlue true. The first step of the proedure is lled to ssign s mny vriles implied y s possile. Also, the onstrint resolvent ( ) is dded s two inry implitions. 3. If hs lredy een ssigned the vlue flse, then is implied nd so is ssigned the vlue flse. The first step of the proedure is lled to ssign s mny vriles implied y s possile. Also, the onstrint resolvent ( ) is dded s two inry implitions. 4. If nd re unssigned, then we hve found new inry luse. 5. If hs lredy een ssigned the vlue true nd hs lredy een ssigned the vlue flse, then is filed literl. Thus is implied. The vrint of Diff used in mrh eq weighs new inry luses tht re produed during the look-hed phse. A ternry luse tht is redued to inry luse tht gets stisfied in the sme itertion of IFIUP, should not e inluded in this omputtion. However, in the urrent implementtion these luses re in ft inluded, whih uses noise in the Diff heuristis. The first step of the IFIUP proedure, omined with the ddition of onstrint resolvents, ensures tht the highest possile mount of vriles re ssigned efore the seond step of the IFIUP proedure. This redues the noise signifintly. An dvntge of IFIUP over generl IUP is tht it will detet onflits fster. Due to the ddition of onstrint resolvents, most onflits will e deteted in the first ll of the first step of IFIUP. In suh se, the seond step of IFIUP is never exeuted. Sine the seond step of IFIUP is onsiderly slower thn the first, n overll speed-up is expeted. Storge of ternry luses in implition rrys requires n equl mount of memory s the ommon lterntive. However, ternry implition rrys llow optimistion of the seond step of the IFIUP proedure. On the other hnd, ternry luses re no longer stored s suh: it is not possile to effiiently verify if they hve lredy een stisfied nd erly detetion of solution is negleted. One knows only tht solution exists if ll vriles hve een ssigned nd no onflit hs ourred. 6 Equivlene Resoning During the pre-proessing phse, mrh eq extrts the so-lled equivlene luses (l 1 l 2 l i ) from the formul nd ples them into seprte dt-struture lled the Conjuntion of Equivlenes (CoE). After extrtion, solution for the CoE is omputed s desried in [7, 13].

8 352 M. Heule et l. In [7] - ppering in the sme volume - we propose new look-hed evlution funtion for enhmrks ontining equivlene luses: let eq n e weight for redued equivlene luse of new length n, C(x) the set of ll redued equivlene luses (Q i ) during look-hed on x, nd B(x) the set of ll newly reted inry luses during the look-hed on x. Using oth sets, the lookhed evlution n e lulted s in eqution (2). Vrile x i with the highest Diff eq (x i ) Diff eq ( x i ) is seleted for rnhing. eq n = n (1) Diff eq = B + Q iǫc eq Qi (2) Besides the look-hed evlution nd the pre-seletion heuristis (disussed in setion 7), the intensity of ommunition etween the CoE- nd CNF-prt of the formul is kept rther low (see figure 4). Nturlly, ll unry luses in ll phses of the solver re exhnged etween oth prts. However, during the solving phse, ll inry equivlenes re removed from the CoE nd trnsformed to the four equivlent inry implitions whih in turn re dded to the implition rrys. The reson for this is twofold: (1) the inry implition struture is fster during the look-hed phse thn the CoE-struture, nd (2) for ll unry luses y i tht re reted in the CoE during IUP(F {x}), onstrint resolvent x y i n e dded to the formul without hving to hek the originl length. We exmined other forms of ommunition, ut only smll gins were notied on only some prolems. Mostly, performne deresed due to higher om- unry luses inry equivlenes CoE pre-seletion heuristis CNF look-hed evlution ommunition during the pre-proessing phse ommunition during the solving phse ommunition during the look-hed phse Fig. 4. Vrious forms of ommunition in mrh eq

9 Mrh eq 353 Tle 2. Performne of mrh eq on severl enhmrks with nd without equivlene resoning without equivlene resoning with equivlene resoning Benhmrks time(s) treesize time(s) treesize speed-up rndom unst 250 (100) rndom unst 350 (100) stnion/hw-n % stnion/hw-n % stnion/hw-n % longmult % longmult % longmult % pyhl-unst % pyhl-unst % qusigroup qusigroup qusigroup zrps/rule dt % zrps/rule dt % munition osts. For instne: ommunition of inry equivlenes from the CNF- to the CoE-prt mkes it possile to sustitute those inry equivlenes in order to redue the totl length of the equivlene luses. This rrely resulted in n overll speed-up. We tried to integrte the equivlene resoning in suh mnner tht it would only e pplied when the performne would enefit from it. Therefore, mrh eq does not perform ny equivlene resoning if no equivlene luses re deteted during the pre-proessing phse (if no CoE exists), mking mrh eq equivlent to its older rother mrh. Tle 2 shows tht the integrtion of equivlene resoning in mrh rrely results in loss of performne: on some enhmrks like the rndom unst nd the qusigroup fmily no performne differene is notied, sine no equivlene luses were deteted. Most fmilies ontining equivlene luses re solved fster due to the integrtion. However, there re some exeptions, like the longmult fmily in the tle. If we ompre the integrtion of equivlene resoning in mrh (whih resulted in mrh eq) with the integrtion in stz (whih resulted in eqstz), we note tht eqstz is muh slower thn stz on enhmrks tht ontin no equivlene luses. While stz 2 solves 100 rndom unst 350 enhmrks ner the treshold on verge in seonds using nodes, eqstz 3 requires on 2 Version t li/englishpge.html 3 Version 2.0 t li/englishpge.html

10 354 M. Heule et l. verge seonds nd nodes to solve the sme set. Note tht no slowdown ours for mrh eq. 7 Pre-seletion Heuristis Overll performne n e gined or lost y performing look-hed on suset of the free vriles in node: gins re hieved y the redution of omputtionl osts, while losses re the result of either the inility of the pre-seletion heuristis (heuristis tht determine the set of vriles to enter the look-hed phse) to selet effetive rnhing vriles or the lk of deteted filed literls. When look-hed is performed on only suset of the vriles, only suset of the filed literls is most likely deteted. Depending on the formul, this ould inrese the size of the DPLL-tree. During our experiments, we used pre-seletion heuristis whih re n pproximtion of our omined look-hed evlution funtion (Ae) [7]. These pre-seletion heuristis re ostly, ut euse they provide ler disrimintion etween the vriles, smll suset of vriles ould e seleted. Experiments with fixed numer of vriles entering the look-hed proedure is shown in % 20% 40% 60% 80% 100% 210 0% 20% 40% 60% 80% 100% () % 20% 40% 60% 80% 100% 50 0% 20% 40% 60% 80% 100% () Fig. 5. Runtime(s) vs. perentge look-hed vriles on single instnes: () rndom unst 350; () longmult10; () pyhl-run-unst ; nd (d) qusigroup () (d)

11 Mrh eq 355 figure 5. The fixed numer is sed on perentge of the originl numer of vriles nd the est vriles (with the highest pre-seletion rnking) re seleted. The plots in this figure do not offer ny indition of whih perentge is required to hieve optiml generl performne: while for some instnes 100% look-hed ppers optiml, others re solved fster using muh smller perentge. Two vrints of mrh eq hve een sumitted to the St 2004 ompetition [11]: one whih selets in every node the est 10 % vriles (mrh eq 010) nd one with full (100%) look-hed (mrh eq 100). Although during our experiments the first vrint solved the most enhmrks, t the ompetition oth vrints solved the sme numer of enhmrks, leit different ones. Figure 5 illustrtes the influene of the numer of vriles entering the look-hed proedure on the overll performne. 8 Tree-Bsed Look-Ahed The struture of our look-hed proedure is sed on the oservtion tht different literls, often entil ertin shred implitions, nd tht we n form shring trees from these reltions, whih in turn my e used to redue the numer of times these implitions hve to e propgted during look-hed. Suppose tht two look-hed literls shre ertin implition. In this simple se, we ould propgte the shred implition first, followed y propgtion of one of the look-hed literls, ktrk the ltter, then propgte the other look-hed literl nd only in the end ktrk to the initil stte. This wy, the shred implition hs een propgted only one. Figure 6 shows this exmple grphilly. The implitions mong, nd form smll tree. Some thought revels tht this proess, when pplied reursively, ould work for ritrry trees. Bsed on this ide, our solver extrts - prior to look-hed - trees from the implitions mong the literls seleted for look-hed, in suh wy tht eh literl ours in extly one tree. The look- 3 4 implition tion propgte 2 propgte 3 ktrk propgte 5 ktrk F 6 ktrk Fig. 6. Grphil form of n implition tree with orresponding tions.

12 356 M. Heule et l. Tle 3. Performne of mrh eq on severl enhmrks with nd without the use of tree-sed look-hed norml look-hed tree-sed look-hed Benhmrks time(s) treesize time(s) treesize speed-up rndom unst 250 (100) % rndom unst 350 (100) % stnion/hw-n % stnion/hw-n % stnion/hw-n % longmult % longmult % longmult % pyhl-unst % pyhl-unst % qusigroup % qusigroup % qusigroup % zrps/rule dt % zrps/rule dt % hed proedure is improved y reursively visiting these trees. Of ourse, the more dense the implition grph, the more possiilities re ville for forming trees, so lol lerning will in mny ses e n importnt tlyst for the effetiveness of this method. Unfortuntely, there re mny wys of extrting trees from grph, so tht eh vertex ours in extly one tree. Lrge trees re oviously desirle, s they imply more shring, s does hving literls with the most impt on the formul ner the root of tree. To this end, we hve developed simple heuristi. More involved methods would proly produe etter results, lthough optimlity in this re ould esily men solving NP-omplete prolems gin. We onsider this n interesting diretion for future reserh. Our heuristi requires list of preditions to e ville, of the reltive mount of propgtions tht eh look-hed literl implies, to e le to onstrut trees tht shre s muh of these s possile. In the se of mrh eq, the pre-seletion heuristi provides us with suh list. The heuristi now trvels this list one, in order of deresing predition, while onstruting trees out of the orresponding literls. It does this y determining for eh literl, if ville, one other look-hed literl tht will eome its prent in some tree. When literl is ssigned prent, this reltionship remins fixed. On the outset, s muh trees re reted s there re look-hed literls, eh onsisting of just the orresponding literl. More speifilly, for eh literl tht it enounters, the heuristi heks whether this literl is implied y ny other look-hed literls tht re the root of some tree. If so, these re lelled hild nodes of the node orresponding to the implied literl. If not lredy enountered, these hild nodes re now reursively

13 Mrh eq 357 d e f g (i) d e f g (ii) d d d f f e f g e g g e (iii) (iv) Fig. 7. Five steps of uilding implition trees (v) heked in the sme mnner. At the sme time, we remove the orresponding elements from the list, so tht eh literl will e heked extly one, nd will reeive position within extly one tree. As n exmple, we show the proess for smll set of look-hed literls. A gry ox denotes the urrent position: Beuse of the order in whih the list is trvelled, literls whih hve reeived higher preditions re lelled s prent nodes s erly s possile. This is importnt, euse it is often possile to extrt mny different trees from n implition grph, nd euse every literl should our in extly one tree. Avilility of implition trees opens up severl possiilities of going eyond resolution. One suh possiility is to detet implied literls. Whenever node hs desendnts tht re omplementry, lerly the orresponding literl is implied. By pproximtion, we detet this for the most importnt literls, s these should hve ended up ner the roots of lrger trees y the ove heuristi. For solvers unle to dedue suh implitions y themselves, we suggest simple, linertime lgorithm tht sns the trees. Some intriguing ides for further reserh hve ourred to us during the development our tree-sed look-hed proedure, ut whih, we hve not een le to pursue due to time onstrints. One possile extension would e to dd vriles tht oth positively nd negtively imply some look-hed literl s full-fledged look-hed vriles. This wy we my disover importnt, ut previously undeteted vriles to perform look-hed on nd possily rnh upon. Beuse of the inherent shring, the overhed will e smller thn without tree-sed look-hed.

14 358 M. Heule et l. Also, one trees hve een reted, we ould inlude non-look-hed literls in the shring, s well s in the heking of implied literls. As for the first, suppose tht literls nd imply some literl. In this se we ould shre not just the propgtion of, ut lso tht of ny other shred implitions of nd. Shring mong tree roots ould e exploited in the sme mnner, with the differene tht in the se of mny shred implitions, we would hve to determine whih trees ould est shre implitions with eh other. In generl, it might e good ide to fous in detil on possiilities of shring. 9 Removl of Intive Cluses The presene of intive luses inreses the omputtionl osts of the proedures performed during the look-hed phse. Two importnt uses n e ppointed: first, the lrger the numer of luses onsidered during the lookhed, the poorer the performne of the he. Seond, if oth tive nd intive luses our in the tive dt-struture during the look-hed, hek is neessry to determine the sttus of every luse. Removl of intive luses from the tive dt-struture prevents these unfvourle effets. When vrile x is ssigned to ertin truth vlue during the solving phse, ll the ternry luses in whih it ours eome intive in the ternry implition rrys: the luses in whih x ours positively eome stisfied, while those luses in wih it ours negtively re redued to inry luses. These inry luses re moved to the inry implition rrys. Tle 4. Performne of mrh eq on severl enhmrks with nd without the removl of intive luses on the hosen pth without removl with removl Benhmrks time(s) treesize time(s) treesize speed-up rndom unst 250 (100) % rndom unst 350 (100) % stnion/hw-n % stnion/hw-n % stnion/hw-n % longmult % longmult % longmult % pyhl-unst % pyhl-unst % qusigroup % qusigroup % qusigroup % zrps/rule dt % zrps/rule dt %

15 Mrh eq 359 Tle 4 shows tht the removl of intive luses during the solving phse is useful on ll kinds of enhmrks. Although the speed-up is only smll on uniform rndom enhmrks, lrger gins re hieved on more strutured instnes. 10 Conlusion Severl tehniques hve een disussed to inrese the solving pilities of look-hed St solver. Some re essentil for solving vrious speifi enhmrks: rnge of fmilies n only e solved using equivlene resoning, nd s we hve seen, mrh eq is le to solve lrge zrps enhmrk y dding only onstrint resolvents. Other proposed tehniques generlly result in performne oost. However, the usefulness of our pre-seletion heuristis is s yet undoutedly sujet to improvement nd will e sujet of future reserh. Referenes 1. D. Mithel, B. Selmon nd H. Levesque, Hrd nd esy distriutions of SAT prolems. Proeedings of AIII-1992 (1992), D. Le Berre, Exploiting the Rel Power of Unit Propgtion Lookhed. In LICS Workshop on Theory nd Applitions of Stisfiility Testing (2001). 3. D. Le Berre nd L. Simon, The essentils of the SAT 03 Competition. Springer- Verlg, Leture Notes in Comput. Si (2004), A. Biere, A. Cimtti, E.M. Clrke, Y. Zhu, Symoli model heking without BDDs. in Pro. Int. Conf. Tools nd Algorithms for the Constrution nd Anlysis of Systems, Springer-Verlg, Leture Notes in Comput. Si (1999), M. Dvis, G. Logemnn, nd D. Lovelnd, A mhine progrm for theorem proving. Communitions of the ACM 5 (1962), O. Duois nd G. Dequen, A kone-serh heuristi for effiient solving of hrd3-st formule. Interntionl Joint Conferene on Artifiil Intelligene (2001), M.J.H. Heule nd H. vn Mren, Aligning CNF- nd Equivlene-Resoning. Appering in the sme volume. 8. O. Kullmnn, Investigting the ehviour of SAT solver on rndom formuls. Sumitted to Annls of Mthemtis nd Artifiil Intelligene (2002). 9. C.M. Li nd Anulgn, Look-Ahed versus Look-Bk for Stisfiility Prolems. Springer-Verlg, Leture Notes in Comput. Si (1997), C.M. Li, Equivlent literl propgtion in the DLL proedure. The Renesse issue on stisfiility (2000). Disrete Appl. Mth. 130 (2003), no. 2, L. Simon, St 04 ompetition homepge. simon/ontest/results/ 12. L. Simon, D. Le Berre, nd E. Hirsh, The SAT 2002 ompetition. Aepted for pulition in Annls of Mthemtis nd Artifiil Intelligene (AMAI) 43 (2005), J.P. Wrners, H. vn Mren, A two phse lgorithm for solving lss of hrd stisfiility prolems. Oper. Res. Lett. 23 (1998), no. 3-5, H. Zhng nd M.E. Stikel, Implementing the Dvis-Putnm Method. SAT2000 (2000),