A Simplifier for Propositional Formulas with Many Binary Clauses

Transcription

1 A Simlifie fo Pooitional Fomula ith Many Binay Claue Ronen I. Bafman Abtact Deciding hethe a ooitional fomula in conjunctive nomal fom i atifiable (SAT) i an NP-comlete oblem. The oblem become linea hen the fomula contain binay claue only. Inteetingly, the eduction to SAT of a numbe of ell-knon and imotant oblem uch a claical AI lanning and automatic tet atten geneation fo cicuit yield fomula containing many binay claue. In thi ae e intoduce and exeiment ith 2-SIMPLIFY, a fomula imlifie tageted at uch oblem. 2-SIMPLIFY contuct the tanitive cloue of the imlication gah coeonding to the binay claue in the fomula and ue thi gah to deduce ne unit liteal. The deduced liteal ae ued to imlify the fomula and udate the gah, and o on, until tabilization. Finally, e ue the gah to contuct an euivalent, imle et of binay claue. Exeimental evaluation of thi imlifie on a numbe of bench-mak fomula oduced by encoding AI lanning oblem ove 2-SIMPLIFY to be a ueful tool in many cicumtance. I. INTRODUCTION PROPOSITIONAL atifiability (SAT) i the oblem of deciding hethe a ooitional fomula in conjunctive nomal fom (CNF) i atifiable. SAT a the fit oblem hon to be NP-comlete [6] and ha imotant actical alication. In the lat decade e have itneed geat oge in SAT olution method, fit ith the intoduction of efficient tochatic local each algoithm [21], [22], and moe ecently ith a numbe of efficient ytematic olve, uch a REL-SAT [4] and SATZ [16] and thei andomized veion [10], and moe ecently, CHAFF[18]. Among AI eeache, inteet in SAT olution algoithm ha inceaed ince Kautz and Selman hoed that ome claical lanning oblem can be olved moe uickly hen they ae educed to SAT oblem [12]. Kautz and Selman lanning a atifiability aoach i baed on geneic SAT technology, and, aide fom the tanlation oce itelf, make no ue of oetie ecific to lanning oblem. Hoeve, SAT-encoded lanning oblem have an imotant yntactic oety: they contain a lage faction of binay claue. Inteetingly, thi oety i found in othe imotant domain automatic tet-atten geneation fo cicuit [15] and bounded model checking [23]. Unlike the geneal SAT oblem, 2-SAT, the oblem of deciding hethe a ooitional fomula containing binay claue only ha a atifying aignment i (contuctively) olvable in linea time. One ould hoe that thi oety ould make it eaie to olve SAT intance containing a lage Autho adde: Deatment of Comute Science, Ben-Guion Univeity, Bee-Sheva, Iael ( bafman@c.bgu.ac.il) faction of binay claue. In thi aticle e decibe the 2- SIMPLIFY eoceo, a imlifie that i geaed to uch fomula. Like othe imlifie (e.g., Cafod COMPACT), thi algoithm take a ooitional fomula in CNF a inut and outut a ne ooitional fomula. Natually, fo thi oce to be othhile, the ne fomula hould be eaie to olve, and the oveall time euied fo imlification and olution of the imlified fomula hould be le than the time euied to olve the oiginal SAT intance. Exeiment on a numbe of bench-mak fomula deived fom lanning oblem ho that 2-SIMPLIFY efomance deend on the difficulty of the oblem, hethe it i atifiable o not, and the olve ued. In many clae of oblem, 2-SIMPLIFY lead to a combined imlification and olution time that i lee than the oiginal olution time. 2-SIMPLIFY efficiently imlement and combine ellknon 2-SAT techniue, a limited fom of hye-eolution, and novel ue of tanitive eduction to educe fomula ize. The baic idea i a follo: each claue of the fom Ô Õ i euivalent to to imlication: Ô Õ and Õ Ô. We ue thi oety to contuct a gah (knon a the imlication gah [1]) fom the et of binay claue. Thi gah contain a node fo each liteal in the language and a diected edge fom a liteal Ð to anothe liteal Ð ¼ if the (dijunction euivalent to the) imlication Ð Ð ¼ aea among the et of binay claue. Afte contucting thi gah, e comute it tanitive cloue and check each liteal to ee hethe it negation aea among it decendant. If Ð i a decendant of Ð, e can immediately conclude that Ð i a coneuence of the oiginal fomula. Once e kno that Ð hold, e can imlify the oiginal fomula. The imlified fomula may contain ne binay claue, hich ae immediately added to the gah. 2- SIMPLIFY utilize thee and othe idea to uickly deive unit liteal fom the oiginal fomula and oduce a imle euivalent fomula a it outut. In the next ection e dicu the backgound of thi ok in moe detail. In Section 3 e eent the imlification algoithm ued by 2-SIMPLIFY. In Section 4 e eent ome exeimental eult, and e conclude in Section 5. II. BACKGROUND We tat ith ome SAT backgound and then e biefly exlain the lanning a atifiability aoach, hich motivated thi ok. A. The SAT Poblem The SAT oblem i defined a follo: given a ooitional fomula in conjunctive nomal fom (CNF) outut YES if the

2 fomula i atifiable and NO otheie. In actice, a oitive ane i accomanied by ome atifying aignment. Thee ae to clae of SAT algoithm: tochatic and ytematic. Stochatic method, uch a G-SAT[21] and WALK- SAT[22], efom tochatic local each in the ace of tuth aignment. Often, they can find olution uickly, but they cannot identify an unatifiable intance. Thei efomance i extemely enitive to the choice of heuitic and vaiou othe aamete. Sytematic method ytematically each the ace of tuth aignment. Thu, they can identify unatifiable intance. Moden ytematic algoithm ae uite fat and table thank to imoved banch choice heuitic and backtacking techniue. In addition, ytematic olve can be imoved by intoducing ome andomization into thei each ocedue, e.g., thei choice of banch vaiable. See [10] fo moe infomation on thi toic. Often, a fomula imlifie i alied befoe the SAT olve. Simlifie ue ecialized, efficient deductive method to educe the oiginal fomula into a imle fomula hich i tyically eaie to olve. The bet knon imlification method i unit oagation. When one of the claue in the fomula contain a ingle liteal, it mut be aigned the value tue in any atifying aignment. Fo examle, if a fomula contain the claue Ô then Ô mut be tue in any atifying aignment, i.e., Ô mut be fale. Once e deduce thi fact, e can ue it to imlify othe claue: claue that contain Ô can be emoved ince thei atifaction i guaanteed hen Ô i fale, and the liteal Ô can be emoved fom any claue containing it (e.g., Ô ill be tanfomed into ) becaue it i euivalent to fale. A e jut a, the imlification oce can yield additional unit claue, hich ae ued to oduce additional imlification. If duing the imlification oce an emty claue i dicoveed (e.g., if e aigned the value tue and thee i a unit claue ) e can conclude that the fomula i unatifiable. Thee ae a numbe of additional imlification method, uch a failed unit liteal and failed binay liteal, hee one o to unit claue ae added to the cuent fomula and e attemt to ho (e.g., uing unit oagation) that the eulting fomula i inconitent. In that cae, the negation of the added claue i imlied by the oiginal fomula, and e udate the tuth aignment accodingly. Fo examle, if ou oiginal fomula become inconitent once e add the claue Ô Õ, e kno that eithe Ô o Õ mut be aigned fale, i.e., that Ô Õ i imlied by the fomula. B. 2-SAT 2-SAT i a ubcla of SAT in hich claue contain no moe than to liteal. While SAT i NP-comlete [6], 2-SAT can be olved in linea time. The key te in olving 2-SAT oblem i the contuction of the imlication gah [1]). The node of the imlication gah coeond to the liteal in the fomula. The gah contain an edge beteen the liteal Ð and the liteal Ð ¼ if the claue Ð Ð ¼ aea in the fomula. That i, edge in the gah coeond to imlication (ince Ð Ð ¼ i euivalent to Ð Ð ¼ ). Since imlication i tanitive, e have that Ð ½ Ð ¾ i imlied by the fomula heneve thee i a ath in the gah ÁÒ Ø Ò % Binay Claue log-di.a 49% log-di.b 55% log-di.c 55% log.d 80% log-g.a 98% log-g.b 98% log-g.c 98% log-un.a 98% log-un.b 98% log-un.c 99% b-di.a 70% b-di.b 71% b-di.c 74% b-di.d 78% TABLE I PERCENTAGE OF BINARY CLAUSES IN SAT-ENCODED PLANNING PROBLEMS beteen node Ð ½ and node Ð ¾. In aticula, if e have a ath beteen Ð ½ and Ð ½, e kno that Ð ½ cannot hold. Theefoe, Ð ½ i imlied by the fomula. If, in addition, e have a ath fom Ð ½ to Ð ½, then neithe Ð ½ no Ð ½ can hold, and o the fomula i unatifiable. Finally, e kno that in evey atifying tuth aignment, if Ð i aigned tue then any liteal imlied by Ð, i.e., any decendant of Ð in the gah, mut be aigned tue a ell. C. Planning A Satifiability The lanning oblem i defined a follo: given a decition of an initial tate, a goal tate, and a et of oeato (=Action) fo changing the tate of the old, find a euence of oeato that, hen alied in the initial tate, yield the goal tate. An imotant develoment in lanning algoithm a Kautz and Selman lanning a atifiability aoach [12]. Kautz and Selman hoed that by educing lanning oblem to atifiability oblem, e can often olve them moe uickly than by uing tandad lanning algoithm. Planning oblem can be encoded a atifiability oblem in a numbe of ay (e.g., ee [8] fo a decition and analyi of ome of thee method). A [5] oint out, encoded lanning oblem contain a lage numbe of binay claue. In Table 1 e ho thi fo a numbe of intance of SAT-encoded lanning oblem. Thi i no accidental henomenon. Cloe inection of the tye of containt exeed ithin encoded lanning oblem make it aaent that many clae of thee containt geneate binay fomula. Fo examle, the containt that if an action i executed at ome time oint then all it econdition mut hold io, oduce binay claue. Similaly, the containt aeting that if an action i executed at ome oint then all it effect mut hold afte the execution yield binay claue a ell. In the encoding ued by the BLACKBOX lanne [13] mutual excluion containt (on action and on tate vaiable) lay a ominent ole. Thee containt ae exeed uing binay claue a ell. Inteetingly, it tun out that the SAT encoding of othe imotant oblem exhibit the ame lage ecentage of binay

3 u u t t u u Fig. 1. The Imlication Gah Fig. 2. Removing Stongly Connected Comonent claue. Thee include tet-atten geneation fo cicuit [15] and bounded model checking [23]. III. THE 2-SIMPLIFY PREPROCESSOR We no exlain the algoithm imlemented by the 2- SIMPLIFY eoceo uing the folloing fomula: Ô Õ Ô Ö Ö Û Ô Ø Ø Ô Ù Ø Ô Ô Õ Õ Ô Ù Û Õ Õ Ö Ú Ñ (1) Contuct Imlication Gah. A gah containing all liteal in the language i contucted ith diected edge fom Ð to Ð ¼ if Ð Ð ¼ i a binay claue. Figue 1 ho the imlication gah fo the fomula above. Note that in the figue, e ue Ô to denote Ô. (2) Collae Stongly Connected Comonent. A ubgah in hich thee i a ath beteen evey ai of node i called a tongly connected comonent (SCC). When a ath fom node Ð to node Ð ¼ exit, e kno that Ð Ð ¼ i a coneuence of ou fomula. Theefoe, all node ithin an SCC imly each othe, and they mut all be aigned the ame value. Once e dicove an SCC e elace it by a ingle node. The childen of thi node ae the childen of the node in the SCC, and the aent of thi node ae the aent of the node in thi SCC. In addition, all liteal in the SCC mut be elaced by the liteal coeonding to thi ne node ithin all non-binay claue. Becaue of the ymmetic natue of the imlication gah, fo evey SCC e dicove, anothe SCC containing the negation of the liteal of thi SCC exit. Thu, heneve e elace an SCC by a ne node labelled by the liteal Ð, e elace the ymmetic SCC by a ne node that i labelled by Ð. In ou examle, the node Ø and Ô fom a tongly connected comonent, and o do thei negation, Ø and Ô. We chooe Ô to eeent the fit SCC and e chooe Ô to eeent the econd SCC. The educed gah i hon in Figue 2. (3) Geneate Tanitive Cloue. No, e geneate the tanitive cloue of the gah. Thi can be done ith one taveal of the (no acyclic) gah in evee toological ode (i.e., by adding to the adjacency lit of each node the childen of it childen). We kno that if Ð ¼ i a child of Ð then Ð Ð ¼ i imlied by the oiginal fomula. We can deduce Ð if eithe: 1) fo ome ooition Ô, both Ô and Ô ae childen of Ð. 2) Ð i a child of Ð. Once e deduce Ð, e can efom unit oagation: all childen of Ð ae aigned the value tue, and e can emove all occuence of Ð fom ithin the claue of ou fomula. If the educed fomula contain ne binay claue, e add the aoiate edge to the gah and udate the tanitive cloue. In Figue 3 e can ee the effect of thi te. Fit, e comute the tanitive cloue of the cuent gah, hon in Figue 3A. In thi gah, e ee that Ù ha Ù a a decendant and that Ô ha Ô a a decendant. Theefoe, e conclude that Ô and Ù mut be aigned the value fale. We can emove node that coeond to aigned ooition (i.e., Ô Ù Ô Ù in ou cae). The eulting gah i hon in Figue 3B. Next, e efom unit oagation, and ou initial tenay claue: Ô Õ Õ Ô Ù Û Õ Õ Ö ae educed to Õ Û Õ Õ Ö The fit claue a emoved becaue it i atified, and a (fale) liteal a emoved fom the next to claue. Since e have ne binay claue, e can udate the gah, a hon in Figue 3C, making ue it i tanitively cloed. In the eulting gah, Û i a child of Û, and e can deduce that Û fale. The educed gah i hon in Figue 3D. (4) Deive Shaed Imlication. Let Ð ½ Ð be ome non-binay claue in the fomula. Let Ä be the et of liteal imlied by Ð fo ½. Let Ä Ä ½ Ä. All liteal in Ä ae coneuence of ou fomula, and e can ue

4 u (B) u (A) (D) (C) Fig. 3. (A) Initial Tanitive Cloue (B) Removal of Aigned Node (C) Udate ith Ne Binay Claue (D) Removal of Aigned Node them to efom unit oagation. Conide the claue Õ Ö, the et of liteal imlied by each of the liteal in thi claue ae Õµ Ö Õµ Õµ, eectively. Thei inteection contain Õ. Hence, e can deduce that Õ i fale. (5) Subumtion Elimination. A claue Ð ½ Ð i ubumed by ¼ н ¼ Ñ if Ñ and ¾ Ð ½ Рм м fo evey ½ Ñ. If ¼ ubume, then the containt ¼ i tonge than. Thu, i edundant. We ue the folloing method to uickly detect hethe one non-binay claue i ubumed by ome binay claue imlied by the imlication gah. Given a claue Ð ½ Ð, e geneate a et Ë containing all the childen of the negation of the liteal in, i.e., the childen of Ð ½ Ð. If any of the liteal in aea in Ë, e kno that i ubumed. (6) Pue-liteal Removal. A ue liteal i a liteal hoe negation doe not occu in any of the claue in the fomula. In that cae, e can aign that liteal the value tue ithout affecting the atifiability of the fomula. (7) Comute Tanitive Reduction. The tanitive eduction of a gah i a gah ¼ ith the ame node a but ith a minimal et of edge uch that a ath beteen Ð and Ð ¼ exit in iff a ath beteen Ð and Ð ¼ exit in ¼. Thu, ¼ i a minimal ub-gah of that maintain node connectivity. We comute the tanitive eduction of the cuent gah in ode to educe the ize of the fomula. Ou imlementation of thi te elie on the fact that e tat ith a tanitively cloed gah. Thu, if e emove fom the lit of childen of a node all of it gandchilden, tating at the oot node and ogeing in toological ode, e obtain a educed gah. (8) Outut Simlified Fomula. We outut a fomula hoe claue conit of the non-binay claue emaining at thi tage, and all binay claue coeonding to edge in the tanitive eduction of the gah. Given the aignment deduced o fa and the maing beteen element of tongly connected comonent, the imlified fomula i euivalent to the oiginal fomula. Fo examle, the outut fo ou oiginal fomula ill be: Ö Ú Ñ togethe ith the atial aignment Ù fale Ô fale Û fale Õ fale. Ste (4) i a novel imlementation of an old techniue (hye-eolution [20]) and te (7) i ne. Both have imotant imact on 2-SIMPLIFY efomance. The Deive Shaed Imlication te enhance the ability of 2-SIMPLIFY to deive unit liteal. In ome cae, it can deive hunded of ne unit liteal uickly. In fact, 2-SIMPLIFY ue a moe ohiticated veion of thi ocedue: if no haed unit liteal exit, e attemt to deive ne binay claue by inteecting the imlication of all liteal but one. Thee binay claue ae then added to the imlication gah. The Comute Tanitive Reduction te lead to a minimal ufficient et of binay claue, leading to malle and imle fomula. We have found thi eduction to have an imotant oitive influence on ytematic olve. IV. EXPERIMENTAL EVALUATION We un extenive tet that examine diffeent aect of 2- SIMPLIFY on a et of bench-mak intance of encoded lanning oblem hich ee ued to tet the REL-SAT olve. In addition, to ee the uefulne of 2-SIMPLIFY on othe oblem e checked it efomance on a hot of veification bench-mak oblem. The exeiment in the fit ection belo, dealing ith encoded lanning oblem ee caied out on a DELL Latitude CP notebook ith a Pentium II-400 oceo ith 64MB RAM unning Linux. Thee oblem ee obtained fom tt ft://ft.eeach.att.com/dit/ai/logitic/ta.z and atlan.data.ta.z. The exeiment in the econd ection, dealing ith diffeent encoded veification oblem

5 Intance Time Aigned+Maed Claue Numbe Ratio b-di.a out of /4675 b-di.b out of /13772 b-di.c out of /50457 b-di.d out of / log-di.a out of /6718 log-di.b out of /7301 log-di.c out of /10719 log.d out of /21991 log-g.a out of /20895 log-g.b out of /29508 log-g.c out of /48920 log-un.a out of /14346 log-un.b out of /21943 log-un.c out of /37121 TABLE II RUNNING TIME AND DEDUCTION POWER OF 2-SIMPLIFY Intance SATZ 2-SIMPLIFY SATZ on 2-SIMPLIFY Total log-di.a log-di.b log-di.c log.d log-g.a log-g.b log-g.c log-un.a log-un.b log-un.c b-di.a b-di.b b-di.c b-di.d TABLE III SOLUTION TIMES FOR SATZ AND 2-SIMPLIFY+SATZ. (hich ee much moe difficult) ee efomed on a PC unning Linux ith a Pentium 4, 180Ghz oceo and a 256KB cache. Thee oblem ee taken fom the benchmak of Miolav Velev at.ece.cmu.edu/ mvelev (the fv and 2dlx intance), Ofe Shtichman (ibm intance) obtainable fom atlib (.atlib.og), and the BMC geneated intance of Biee, Cimatti, Clak, and Zhu, -2.c.cmu.edu/ modelcheck/bmc/bmcbenchmak.html. 2-SIMPLIFY i itten in C++ and all time meauement efe to CPU time. A. Encoded Planning Poblem Fit, e examined 2-SIMPLIFY ability to deduce unit liteal. In Table 2 e ho 2-SIMPLIFY unning time on each of the intance, the numbe of vaiable it a able to aign o ma, and the atio beteen the numbe of claue in the imlified fomula and the oiginal fomula. To ae the utility of 2-SIMPLIFY e geneated imlified fomula fo each of the intance and comaed the olution time of the oiginal fomula ith the combined imlification and olution time fo the imlified fomula. We efomed thi comaion uing to ytematic olve: SATZ and REL- SAT. The eult fo SATZ ae hon in Table 3. In 5 out of the 14 oblem, the ue of 2-SIMPLIFY lead to degaded efomance. Thi occu on the elatively malle oblem, and tyically, ith a mall ovehead that tem fom imlification cot. On 9 out of the 14 oblem, 2-SIMPLIFY lead to imoved efomance. Of aticula imotance i the fact that 2- SIMPLIFY efom bette on the oblem that hade fo SATZ and on thoe that SATZ cannot olve ithout imlification. The eult fo REL-SAT ae hon in Table 4. The imlified fomula ee olved uing REL-SAT, but ith it eoceo diabled. In many cae, thi lead to imoved efomance, a natual coneuence of the fact that 2-SIMPLIFY ovide tonge imlification caabilitie. Fo examle, in the laget intance, b-di.d, REL-SAT ithout the eoceo took 571 econd, intead of 847. Hoeve, thee ae cae in hich the REL-SAT eoceo lead to bette unning time on the imlified fomula. Fo intance, in the log-g and log-un intance, it a alay bette. Oveall, e ee that Intance REL-SAT 2-SIMPLIFY 2-SIMPLIFY+REL-SAT log-di.a log-di.b log-di.c log.d log-g.a log-g.b log-g.c log-un.a log-un.b log-un.c b-di.a b-di.b b-di.c b-di.d TABLE IV SOLUTION TIMES FOR REL-SAT, 2-SIMPLIFY, AND 2-SIMPLIFY+REL-SAT. 2-SIMPLIFY+REL-SAT i almot alay fate than REL-SAT alone, ith thee excetion that tem fom elatively long imlification time. The imovement i eecially ignificant in the hadet intance. We note that the REL-SAT figue eeent aveage unning time (becaue REL-SAT ha a tochatic element) and that in the cae of b-di.d, REL-SAT timed out on the oiginal oblem in ome of the iteation and the eult ovided i a loe bound on it tue aveage unning time. We un anothe euence of exeiment to comae 2- SIMPLIFY ith Cafod COMPACT imlifie. COMPACT ovide vaiou imlification otion. The baic COMPACT imlifie efom unit eolution, emove atified claue, and ename vaiable to be contiguou. In addition, thee ae a numbe of otional flag. With the Ô flag, COMPACT efom ue-liteal elimination, ith it eolve aay liteal that occu only once, ith Ð it efom the unit-failed tet that i, it check fo each liteal hethe adding thi liteal to the fomula lead to an inconitency (uing unit oagation). If o, it aign that liteal the value fale. Finally, the otion add the binay-failed tet. In thi cae, ai of liteal ae added each time, and unit eolution i efomed. If an inconitency i detected, the negation of the conjunction of thi ai of liteal

6 ÁÒ Ø Ò COMPACT+SATZ 2-SIMPLIFY+ SATZ l + SATZ b-di.a b-di.b b-di.c b-di.d log-di.a log-di.b log-di.c log.d log-g.a log-g.b log-g.c log-un.a log-un.b log-un.c TABLE V SIMPLIFICATION WITH COMPACT AND 2-SIMPLIFY, SOLUTION WITH SATZ. ÁÒ Ø Ò COMPACT+REL-SAT 2-SIM.+REL-SAT l + REL-SAT b-di.a b-di.b b-di.c b-di.d log-di.a log-di.b log-di.c log.d log-g.a log-g.b log-g.c log-un.a log-un.b log-un.c TABLE VI SIMPLIFICATION WITH COMPACT AND 2-SIMPLIFY, SOLUTION WITH REL-SAT. i added to the fomula. Thi latte tet i uite oeful, but it i almot alay too cotly to be othhile. Afte teting vaiou combination of otion on the above intance, e found that the bet efomance i obtained almot alay uing eithe no flag o uing the Ô Ð flag, and thi i hat e ho hee. In Table 5 and 6 belo, e comae the imlification + unning time of SATZ and REL-SAT on COMPACT imlified fomula and on the 2-SIMPLIFY imlified fomula. In Table 5, e ee the eult fo SATZ. The column coeond to the combined unning time of SATZ and the imlification algoithm on each of the intance. Thee ae thee cae in hich COMPACT ith no otion lead to bette efomance, but oveall, 2-SIMPLIFY lead to much bette eult, eecially on the moe difficult intance. The eult hen the Ô Ð otion i ued ae moe vaied. 2-SIMPLIFY till doe bette in moe cae, but on the lat to intance, COMPACT ith Ô Ð doe much bette. Thi may indicate that an enhanced veion of 2- SIMPLIFY ith Ô Ð-like caabilitie could efom bette than eithe imlifie. Hoeve, a e hall ee belo, the elative efomance i geatly affected by the olve ued. In Table 6 e ho the coeonding eult fo REL-SAT (ithout it eoceo). Again, e ee that 2- SIMPLIFY i bette than COMPACT ith no otion, and that 2-SIMPLIFY and Ô Ð ucceed on diffeent intance. Hoeve, notice that Ô Ð doe noticeably bette only on 3 intance. Moeove, notice the lage change in efomance on the to oblem that ee mot difficult fo SATZ log-un.b/c. Again, it ould be excellent if e could get the bet of both old. Finally, e examined 2-SIMPLIFY influence on the efomance of WALKSAT, a tochatic olve. A noted, tochatic olve euie tuning, and e tied to find the bet aamete in each cae. A Table 7 ho, the eult ae, again, uite oitive. Although thee ae thee intance in hich 2- SIMPLIFY lead to educed efomance, on mot intance it lead to imoved efomance. Moeove, on all the hade intance, 2- SIMPLIFY lead to noticeable imovement. Intance WALKSAT 2-SIMPLIFY+WALKSAT b-di.a b-di.b b-di.c b-di.d log-di.a log-di.b log-di.c log.d log-g.a log-g.b log-g.c B. Veification Poblem TABLE VII SOLUTION TIMES USING WALKSAT. A natual uetion i hethe the efomance of 2- SIMPLIFY caie ove to othe SAT-encoded oblem. In aticula, thoe oiginating fom the veification community. To check thi, e teted 2-SIMPLIFY on a hot of model-checking and bounded model-checking oblem. Thee oblem ae much lage than the lanning bench-mak, and SATZ and REL- SAT have a vey difficult time olving them. Luckily, a ecent olve, CHAFF, i able to olve thee oblem athe eaily. Thu, e comaed the unning time of CHAFF ith and ithout 2-SIMPLIFY on thee oblem. In Table 8 e ee the efomance of 2-SIMPLIFY on atifiable intance. CHAFF olve thee oblem ithout difficulty, uually in le than a econd. Although CHAFF ok fate on the imlified oblem, the imlification time i oughly 40 econd, making the ue 2-SIMPLIFY inaoiate. In Table 9, e ee the eult on a et of unatifiable oblem. Hee, e ee eciely the ooite ictue. Excet fo to cae, on all intance that euie moe than 10 econd, 2-SIMPLIFY lead to conideable eduction in combined unning time. The hadet oblem in thi cla, fv-7ie, a not olvable by CHAFF ithin ove 1000 minute, heea it imlified veion a olved ithin le than an hou.

7 Intance CHAFF CHAFF on imlified 2-SIMPLIFY 2-SIMPLIFY+ CHAFF 2dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug dlx...f2-bug ibm OOM ibm ibm ibm TABLE VIII SOLUTION TIMES FOR CHAFF ON ORIGINAL AND SIMPLIFIED SATISFIABLE PROBLEMS(OOM STANDS FOR OUT OF MEMORY). The exlanation fo the efomance diffeence of 2- SIMPLIFY ith eect to the cla of atifiable and unatifiable intance mot likely lie in the diffeent natue of the each euied. CHAFF eem to be able to geneate a ingle olution, hen one exit, uickly. Thi make the advantage of the eduction elatively mall even if e ave duing olution time, the diffeence i too mall to comenate fo the imlification cot. On the othe hand, to ove that a oblem i unatifiable, an exhautive each of the ace of aignment i euied. No, the malle each ace of the imlified fomula ay off. V. CONCLUSION AND RELATED WORK SAT intance ith many binay claue aie natually in a numbe of imotant alication. The abundance of binay claue in uch oblem can be exloited uing 2-SAT olution method and othe ecialized infeence algoithm. Hee, e eented 2-SIMPLIFY, a inciled and efficient imlification algoithm that ue the tanitive cloue of the imlication gah togethe ith a novel imlementation of hye-eolution (i.e, the deive haed imlication te) and tanitive eduction to obtain a malle euivalent fomula. Thi lead to an aoach that i fate, moe oeful, and moe efficient than the ad-hoc eolution of binay claue ued in [5]. Ou exeiment ho that the efomance of 2- SIMPLIFY deend cucially on the olve ued, the atifiability of the oblem, and it difficulty fo the olve. Such iegula behavio of imlifie a noticed by [17]. Hoeve, it eem afe to ay that on oblem that ae difficult fo a olve, 2-SIMPLIFY i uite ueful. Indeed, e a that on the encoded lanning oblem 2-SIMPLIFY a beneficial in conjunction ith both SATZ and REL-SAT in the bulk of cae. On the lage atifiable veification oblem e a that 2-SIMPLIFY imlification time a much lage than the olution time. Hoeve, on the lage unatifiable veification time, e a that 2-SIMPLIFY lead to to to thee-fold imovement, and even moe. Anothe inteeting obevation i the comlementay effect of COMPACT Ô Ð otion and 2-SIMPLIFY, aiing the natual uetion of hethe the caabilitie of thee to imlifie can be combined effectively. Finally, e note that the cuent imlementation of 2-SIMPLIFY leave much fo imovement, and e believe that a moe caeful deign can lead to much imoved imlification time. Thi i of aticula imotance hen one ecall that in almot all cae the imlified fomula euie le time to olve. Thu, educed efomance i often due to the imlification time ovehead. We ae not the fit to utilize binay eolution in thi aea. Laabee ued the imlication gah to devie a SAT algoithm in the context of tet-atten geneation [15]. Laabee ytematically geneate atifying aignment conitent ith the imlication gah. Any aignment that atifie the non-binay claue i a atifying aignment fo the hole fomula. Thi method exloit the binay otion of the fomula, but it doe not utilize the oe of contemoay vaiable odeing and each techniue. 2CL [9] i a olve baed on the Davi-Putnam-Logemann- Loveland algoithm [7]. At each banch oint, 2CL contuct the tanitive-cloue of the cuent imlication gah and ue it to chooe the next banching vaiable. Thu, 2CL i a dynamic extenion of a key aect of 2-SIMPLIFY. It doe not incooate ou deive haed imlication te. We did not exeiment ith 2CL but cuently, it i not conideed a cometitive olve. Indeed, extending 2-SIMPLIFY to a full olve ould eem to be a natual next te. Thi olve ill be baed on the DPLL algoithm. Evey time an aignment i made, the imlication gah ill be ued to detect all of it immediate imlication. Ne binay claue eulting fom the eduction of tenay claue ill be added to the imlication gah, and the deive haed imlication te ill be executed. Moeove, the imlication gah may be able to ovide u ith valuable infomation fo banch election. Unfotunately, ou initial effot in thi diection ee not ucceful. Thee aea to be to eaon fo thi. Fit, in ou imlementation, maintaining the gah tanitively cloed afte each aignment (hich tyically eult in a numbe of ne binay claue) aeaed to be a eiou bottleneck. Second, e ee not able to come u ith a uick, yet oeful banch election heuitic that i cometitive ith that ued by cuent olve. Fo intance, it aea that SATZ i able to gain

8 Intance CHAFF CHAFF on imlified 2-SIMPLIFY 2-SIMPLIFY+ CHAFF bael bael ueueinva ueueinva ueueinva fv-3ie fv-3ie fv-3ie fv-3ie fv-4ie fv-4ie fv-4ie fv-4ie fv-4ie fv-5ie fv-5ie-ooo fv-6ie fv-6ie-ooo fv-7ie 60, TABLE IX SOLUTION TIMES FOR CHAFF ON ORIGINAL AND SIMPLIFIED UNSATISFIABILITY PROBLEMS. much infomation uickly uing it unit oagation te, hile e have not been able to emulate that uing the infomation in ou imlication gah. Vey ecently, Bacchu eoted on hi effot to extend ou aoach to a full olve [2]. The eulting olve i cometitive ith CHAFF on ome SAT intance. In aticula, Bacchu ugget that eeated alication of ou deive haed imlication te can be vey ueful. Ou ue of gah-baed techniue, motivated by ell-knon 2-SAT techniue i omehat eminicent of gah-baed imlification techniue ued in ok on automated theoem oving uch a Koalki Claual Gah [14] hich fomed the bai of the Makgaf Kal Refutation Pocedue [19] (the imlication gah can be vieed a a ecial cae of thi gah), and ome eite-baed imlification method [11], [3]. Hoeve, the method ued thee ae much moe ohiticated and focu on aect of fit-ode theoie that do not come u in the ooitional etting in hich e ok. Acknoledgment: I am gateful to Yefim Dinitz and Avaham Melkman fo thei hel and advice on gah algoithm and fo imotant comment on eviou veion of thi ae, and to the anonymou eviee fo thei ueful uggetion and comment. Thi ok a uoted in at by the Paul Ivanie Cente fo Robotic and Poduction Management. REFERENCES [1] B. Avall, M. Pla, and R. Tajan. A linea-time algoithm fo teting the tuth of cetain uantified boolean fomula. Infomation Poceing Lette, 8: , [2] F. Bacchu. Enhancing davi utnam ith extended binay claue eaoning. In Poc. AAAI 02, [3] L. Bachmai and H. Ganzinge. Reite-baed euational theoem oving ith election and imlification. Jounal of Logic and Comutation, 4(3): , [4] R. J. Bayado and R. C. Schag. Uing CSP look-back techniue to olve eal-old SAT intance. In Poc. AAAI-97, age , [5] R. I. Bafman. Reachability, elevance, eolution, and the lanning a atifiability aoach. In IJCAI 99, age , [6] S. A. Cook. The comlexity of theoem oving ocedue. In Poc. of the 3d ACM Symoium on Theoy of Comuting. ACM, [7] M. Davi, G. Logemann, and D. Loveland. A machine ogam fo theoem oving. Communication of the ACM, 5(7): , July [8] M. D. Ent, T. D. Milltein, and D. S. Weld. Automatic SAT-comilation of lanning oblem. In Poceeding of the Intenational Joint Confeence on Atificial Intelligence, [9] A. Van Gelde and Y. K. Tuji. Satifiability teting ith moe eaoning and le gueing. In D. S. Johnon and M. Tick, edito, Cliue, Coloing, and Satifiability: Second DIMACS Imlementation Challenge. Ameican Mathematical Society, [10] C. P. Gome, B. Selman, and H. Kautz. Booting combinatoial each though andomization. In Poc. of 15th Nat. Conf. AI, age , [11] D. Kau and H. Zhang. An ovevie of eite ule laboatoy (l). In Poc. of RTA 89, Lectue Note in Comute Science 355, age Singe-Velag, [12] H. Kautz and B. Selman. Puhing the enveloe: Planning, ooitional logic, and tochatic each. In Poc. of the 13th National Confeence on AI (AAAI 96), age , [13] H. Kautz and B. Selman. Unifying at-baed and gah-baed lanning. In Poc. 16th Intl. Joint Conf. on AI (IJCAI 99), age , [14] R. Koalki. A oof ocedue uing connection gah. Jounal of the ACM, 22: , [15] T. Laabee. Tet atten geneation uing boolean atifiability. IEEE Tanaction on Comute-Aided Deign, age 4 15, Januay [16] C. M. Li and Anbulagan. Heuitic baed on unit oagation fo atifiability oblem. In Poc. IJCAI-97, [17] I. Lynce and J. P. Maue-Silva. The inteaction beteen imlification and each in ooitional atifiability. In Poc. of CP 01 Wokho on Modeling and Poblem Fomulation, [18] M. Mokeicz, C. Madigan, Y. Zhao, L. Zhang, and S. Malik. Chaff: Engineeing an efficient at olve. In Poc. of 39th Deign Automation Confeence, [19] Han Jügen Ohlbach and Jög H. Siekmann. The Makgaf Kal efutation ocedue. In J.-L. Laez and G. Plotkin, edito, Comutational Logic, Eay in Hono of Alan Robinon, age MIT Pe, [20] J. A. Robinon. Automatic deduction ith hye-eolution. Int. J. of Com. Math, 1: , [21] B. Selman, H. J. Leveue, and D. Mitchell. Gat: A ne method fo olving had atifiability oblem. In Poc. of the 10th National Conf. on AI (AAAI 92), age , [22] Bat Selman, Heny A. Kautz, and B. Cohen. Noie tategie fo imoving local each. In Poc. Nat. Conf. on AI, age , [23] O. Shtichman. Tuning at checke fo bounded model checking. In E.A. Emeon and A.P. Sitla, edito, Comute Aided Veification 2000, 2000.