Procedure di Decisione Ibride per QBF Hybrid Evaluation Procedures for QBF

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1 Procedure di Decisione Ibride per QBF Hybrid Evlution Procedures for QBF Mrco Benedetti Istituto per l Ricerc Scientific e Tecnologic (IRST) Vi Sommrive 18, Povo, Trento, Itly benedetti@itc.it

2 Abstrct In questo lvoro viene presentto un sistem per l vlutzione e certificzione di formule proposizionli quntificte (QBFs) che si bs su un pproccio ibrido l problem. Vengono considerte questioni ttinenti ll integrzione di differenti filosofie di soluzione e il ruolo delle met-euristiche nell composizione del sistem complessivo. We present system designed to evlute nd certify Quntified Boolen Formuls (QBFs) by employing hybrid pproch to the problem. Issues in integrting different philosophies nd the usge of met-heuristics to compose the whole frmework re ddressed. Keywords: Propositionl Resoning, QBF, Hybrid lgorithms, Symbolic resoning. 1 Introduction We present n pproch to the hybridiztion of severl different evlution lgorithms for Quntified Boolen Formuls (QBFs). A QBF is propositionl formul in which toms cn be existentilly or universlly quntified. The evlution of QBF instnces is decidble, PSPACEcomplete problem, nd plenty of pplictions exist for this logic formlism: Every problem tht cn be stted s two-plyer finite gme cn be modeled in QBF. Mny rel-world pplictions re two-plyer gmes: Unbounded model checking for finite-stte systems [28] nd conformnt plnning [27] just to nme two relevnt exmples hve hndy QBF formultions. Hence, QBF is regrded s promising formlism, but substntil improvements in decision procedures re expected before its potentil cn be unleshed to pplictions [1, 21, 8]. Severl different prdigms hve been proposed nd implemented to evlute QBF formuls, ech one feturing specific strengths nd weknesses. No one domintes the others, nd ttempts to mix such inference styles with one nother hve not been undertken so fr. We now briefly describe the mjor pproches to QBF evlution, nd discuss some of their distinguishing fetures. DPLL-like solvers. Serch-bsed solvers extend the DPLL-pproch [12] to the quntified cse [10]. Models re serched for in the most nturl wy: Following the left-to-right order of the vribles in the prefix during topdown, depth-first visit of the semntic evlution tree of the formul. This is the most common pproch to QBF evlution. Exmples of solvers in this clss re Quffle[31], QuBE [16], nd semprop [22]. Resolution-bsed solvers. Rther thn serch for model, it is possible to solve the formul by pplying refuttionlly complete inference procedure (quntor [8], QMRES [26], QBDD [26]). Such strtegy ims to derive necessry consequences from the given formul, ending up with the empty cluse if nd only if the originl formul is unstisfible. These methods build upon generliztions of the resolution pproch to existentil stisfibility, such s q-resolution [19, 9]. Skolemiztion-bsed solvers. Skolemiztion-bsed solvers replce the originl QBF evlution problem with the vlidity problem of the skolemized instnce (existentil quntifiers re eliminted by replcing the vribles they bind with Skolem functions whose definition domins re ppositely chosen to preserve stisfibility.). Once skolemiztion is pplied, methods from F OL utomted theorem proving or d-hoc strtegies such s the ones presented in [2, 5] cn be employed. SAT-bsed solvers. The mening of QBF instnce cn be prtilly or totlly [2, 5] expnded into purely existentil SAT instnce, which is then ddressed by mens of stte-of-the-rt SAT solvers.

3 Symbolic solvers. Such solvers employ BDD-bsed or ZDD-bsed compressed representtion for cluse sets, in the spirit of [11], then pply either serch (ZQSAT[14]) or resolution (QMRES [26]). 2 Specific Fetures nd Issues in Integrtion DPLL-bsed solvers inherit the experience of decde of successful ppliction of similr techniques to the purely existentil cse (SAT solvers). Mny lookbck enhncements such us conflict nlysis, filure-driven ssertion, non-chronologicl bcktrcking, nd lerning nd lookhed enhncements such us forwrd checking, forwrd resoning nd heuristics choices hve been pplied over the yers (see [18, 13, 29, 30, 17, 23, 24] for exmples nd [4] for survey). In ddition to their long history, DPLL-like techniques re ttrctive s they gurntee to work in polynomil spce, which is relly good point when ll the other pproch fil. However, their brnching behvior integrtes hrdly with other techniques, nd it is often bsed on purposely designed dt structures (e.g.: wtched literls) with strong trdeoff between efficiency nd generlity. SAT-bsed QBF solvers fce difficult issue: By reducing PSPACE-complete problem to n NP-complete problem, we cuse in generl n exponentil blowup in the size of ech instnce. For rel-world problems, this mens 10 x, x > 100 cluses: Definitely unffordble instnces re thus generted. Nevertheless, the clss of lmost-existentil problems (very few universl quntifiers) comes out to be menble to n extremely efficient SAT-bsed solution. To minor extent, resolution-bsed solvers lso suffer from spce-explosion problems. Intermedite clusesets much lrger thn the originting instnce re possibly generted to solve the problem. Out-of-memory conditions re likely to occur, nd the solver hs to simply give up the solution process (provided it is ble to detect tht physicl memory is over: n OS trshing condition occurs otherwise). Integrtion is mde difficult by enormous oscilltions in the instnce size. On the positive side, resolution-bsed solvers pply somehow complementry pproch to QBF evlution w.r.t serch bsed solvers: The ltter consider quntifiers in outermost-to-innermost direction, the former employ the opposite strtegy, using q-resolution [19] for existentil quntifiers nd expnsion for universl ones. When combined with ( lot of) other mechnisms (equivlence resoning, subsumption control, estimtion of expnsion nd resolution costs, scheduling heuristics) resolution-bsed pproches yield the best solvers overll on lrge fmily of instnces [20]. The so-clled symbolic pproches show certin strength on specific clsses of instnces, but seem to be not competitive in generl [25] for the SAT cse. Things chnge lot in the QBF scenrio. The ide of compressed/symbolic representtions seem to be very promising [26, 8, 14] s fr s QBF is concerned. Possible resons for this symmetry discussed in the literture re (1) tht symbolic representtions mnge existentil nd universl quntifiers in n lmost-symmetric wy, wheres serch-bsed procedures hve hrd wy with the ltter ones, nd (2) tht QBF problems re more structured nd less combintoril in nture thn their propositionl counterprts. Skolemiztion-bsed solvers for QBF hve been implemented recently[5]. More thn in specific inference style, the successful feture of such solvers lys in their structurl bility to move the problem to different representtion spce: Those of the definbility of the skolem terms. Besides immeditely llowing for the ppliction of number of simplifiction techniques, the spce of the skolemized instnces seem to be suited for the ppliction of mny of the bove inference styles. This is the reson why the hybrid rchitecture we present hs skolemiztion t its hert. 3 A Hybrid System skizzo [2, 3] is softwre suite for deling with QBFs, minly imed t evluting prenex CNF formuls by mens of novel symbolic skolemistion technique[5] on top of which numerous inference styles re mixed. In ddition, it enbles the user to (A) experiment with quntifier trees[7], (B) certify the (un)stisfibility of formuls[6] nd (possibly) extrct unstisfible cores, nd (C) compute, mnge, nd query stnd-lone certifictes of stisfibility for QBFs. At the herth of skizzo stys new kind of symbolic representtion for cluses nd formuls, bsed on Binry Decision Digrms (BDDs). As opposed to previous BDD-bsed pproches to propositionl logic, skizzo s one employs two-level dt structure [2] designed to tke dvntge of the distinguishing fetures of QBFs. Besides llowing for novel style of (complete/incomplete) symbolic resoning, such representtion mkes it possible to unify within coherent frmework mny other pproches to QBF-stisfibility implemented so fr. 4 Representtion of QBF Instnces nd Dt Structures Three representtion spces for QBFs coexist within skizzo. They re interconnected by two stisfibilitypreserving trsformtions (pplied one-wy), s reported in the picture below. The first trnsformtion leverges outer skolemistion to mp ny (prenex CNF) instnce F QBF s onto symbolic formul F = SymbSk(F ), which is sid to be symbolic s it couples list-bsed nd BDD-bsed dt structures to compctly represent (possibly) exponentilly less succinct propositionl formul. The sentence F encodes the definbility of set of Skolem functions tht cpture model (if ny) of the originl instnce, ccording to the symbolic skolemistion technique

4 presented in [5]. A forml semntics is ssocited to symbolic formuls in so s F st SymbSk(F ) for every F. The other trnsformtion clled groundistion trnsltes symbolic formul F into purely existentil CNF propositionl instnce P rop(f) ( SAT problem) such tht F st SymbSk(F ) st P rop(symbsk(f )). The role of these representtions is s follows: Plin QBFs re hndled in pre-processing phse. Then, skizzo moves to the symbolic representtion nd performs most of its work thereon. Zero or more CNF instnces re generted/solved during the whole process. Symbolic skolemistion (nd most of the processes described below) relies on the existence of quntifier tree stting which existentil vribles re in the scope of which universl vribles. Such tree-shped structures re extrcted out of the flt prenex input ccording to [7]. They replce liner prefixes so to more closely reflect the intrinsic dependencies in the mtrix. Prenex QBFs (CNF) symbolic skolemistion Symbolic Formuls groundistion Prop. CNF instnces A smple quntifier tree for the QBF b c d e f g h.( c) ( h) (c d g) ( b f) ( c e h) ( b f) ( c g) is depicted in Figure 1. The symbolic representtion is designed to llow for efficient forms of symbolic resoning (Section 2), where universl resoning is tken prt form existentil resoning (ROBDDs conveniently del with the former, list-bsed bf f [h] 0 1 c e [c, h] d e 0 1 Figure 1: A smple quntifier tree representtions with the ltter). A symbolic formul is mde up by symbolic cluses. During symbolic skolemistion, one symbolic cluse is extrcted out of ech QBF cluse. The two mjor components of symbolic cluse Γ I re list Γ of existentil literls nd n index-set I represented vi ROBDD whose support set is the set of universl vribles dominting the existentil node t which the cluse is ttched in the quntifier tree. For exmple, the symbolic cluses [h] {00,01} nd [c, h] {10} re extrcted out of h nd c e h respectively (see the picture). Ech symbolic cluse Γ I compctly represents set P rop(γ I ) (with crdinlity I ) of ground propositionl cluses, in such wy tht F is st iff P rop(f) is st. For exmple, P rop([c, g] {01,10} ) = {c 0 g 01, c 1 g 10 }. The symbolic size of F is F, its ground size is P rop(f) : The initil symbolic size of F is thus liner in F (see [5]). 5 Inference Strtegy The inference strtegy followed by skizzo chnges s the solution process goes on. Its evolution is described by finite stte mchine (Figure 2) whose inference sttes re S inf = {G, S, R, B, G}. Ech stte in S inf is ssocited to the ppliction of n inference style. Ech trnsition x y in Figure 2 (x, y S inf ) is lbeled by condition tht triggers the shift from the style x to y (possibly requiring stisfibility-preserving trnsformtion). We now briefly describe ech stte nd trnsition in turn. Q: Ground QBF Resoning. In the Q-stte skizzo works in the originl QBF spce, s represented below. The step Q 1 mounts to pply Q: Ground QBF Resoning the quntified form of three Q simple (incomplete) inference 1 : Normliztion rules: unit cluse propgtion, Q pure literl elimintion, nd 2 : Tree Reconstruction forll-reduction. The trnsition Q 1 Q 2 is triggered when ll Q 3 : Bounded Vr. these rules rech their fixpoint. Elimintion Bounded vrible elimintion (Q 3 ) pplies q-resolution to S eliminte selected existentilly quntified vrible v in the deepest existentil scope of some brnch of the quntifier tree. This is done by substituting ll the cluses contining v with the set of ll the resolvents over v. As repeted pplictions of vrible elimintion often led to n unmngeble explosion of the number of cluses, bounded form of elimintion is employed: Only vribles whose elimintion shrink the overll number of literls or cluses re eligible for elimintion. The trnsition Q 3 Q 1 is selected when t lest one vrible hs been eliminted during the lst round, Q 3 S is followed otherwise. S: Incomplete Symbolic Resoning. The instnce is ttcked by mens of set of (incomplete) symbolic in-

5 Q: Ground QBF Resoning symbolic skolemistion S: Incomplete Symbolic Resoning R: Complete Symbolic Resoning B: Symbolic Brnching Resoning groundistion G: SAT-bsed CNF Resoning Figure 2: Inference FSM ference rules, designed fter their ground counterprts to chieve in one single ppliction on symbolic cluses the sme result they would obtin if pplied to ech ground cluse seprtely. SUCP (Symbolic Unit Cluse Propgtion). This rule builds on top of the observtion tht ech symbolic unit cluse [γ] I in the formul represents set {γ i i I} of ground unit literls. All of them re symboliclly ssigned t once to void n immedite contrdiction. SPLE (Symbolic Pure Literl Elimintion). This rule computes symbolic representtion for the set of pure literls, then simplify the formul by ssigning ll of them t once. It comes in two flvors: monolitic (one vrible per step) nd n incrementl (one cluse per step) version. SSUB (Symbolic SUBsumption). This rules removes ll the symbolic cluses tht re subsumed by other cluses (forwrd subsumption). It employs scheduling heuristics, lzy computtions, nd signturebsed mechnism to minimize the overll effort. This rule complements the bckwrd subsumption mechnism which is pplied on-the-fly t ech cluse insertion. SHBR (Symbolic Hyper Binry Resolution). This rules enumertes ll the resolution chins of binry symbolic cluses in the formul, looking for contrdictions. Ech such contrdiction determines necessry consequence of the formul, compctly represented s unit symbolic cluse which is dded to the instnce (SUCP then drws ll the entiled consequences). SER (Symbolic Equivlency Resoning). This rules look for non-empty strongly connected components in the symbolic binry impliction grph[2] of the formul. Ech such component determines symbolic equivlence which is pplied to simplify the formul. A crefully designed ppliction schedule is necessry to profit from the bove set of rules s whole. skizzo implements dynmic scheduling policy which works s follows. 1. The inference process is divided into subsequent inference rounds. At ech round, the rules tht hve the rights to do so (see below) re sequentilly executed. 2. The rule currently working is monitored during its execution. When certin resource limits re exceeded (inference steps undertken, time elpsed, memory llocted, etc.), the rule is preemptively stopped (the rule s context is sved to re-strt working from the interruption point). 3. When ll the rules in the inference round hve been executed, they re rnked ccording to their reltive efficiency. The resource limits for the next rounds re re-distributed on meritocrtic bsis: the better rule hs proved to be, the lrger the resources it will be grnted next. 4. In ddition, rules filing to be effective loose the right to execute for number of inference steps tht enlrges with the number of rounds they hve been performing poorly. The longer they keep on being ineffective, the more springly they re given try. The ssessment of rules efficiency is mjor issue in the bove policy. As ll the rules reduce the ground size of the instnce t ech ppliction (conversely, the symbolic size might be enlrged), the ground-size-shrinkpercentge-per-resource-unit is ssumed s mesure of efficiency. This mesure needs itself resources to be computed. When BDD primitives nd lzy evlution do not suffice to keep the cost of ssessment within preestblished limits, skizzo resorts to pproximted mesures. The trnsition S G is triggered if the ground size of the current problem becomes ffordble vi SAT-bsed resoning (see the G-style), unless the symbolic resoning is behving so efficiently tht ground resoning is estimted not to py bck. The trnsition S R is ctivted when the rules dopted come out to be unble to solve the problem. This hppens under two circumstnces: (1) the overll fixpoint is reched but no decision is obtined, or (2) the rte t which the problem is being shrunk hs been stying below certin threshold since given number of inference rounds. R: Complete Symbolic Resoning. This stte is similr to S, with one mjor exception: refuttionlly complete rule is inserted in the pool of symbolic rules exercised t ech inference round. SDR (Symbolic Directionl Resolution). This rules elimintes one symbolic vrible per step by substituting

6 the set of resolving cluses with the set of their symboliclly computed resolvents. Efficiency s size-shrinking mesurement is unfir for SDR. This rule my need to pss through intermedite cluse-sets tht re much lrger thn the originting instnce to come to solution. So, SDR is given the chnge to consume more nd more resources regrdless of the size of the formul it is constructing. The other rules re still pplied/evluted in round robin wy (SSUB is especilly useful here to reduce the redundncy SDR genertes). Two outcomes re possible: (1) the lrgest intermedite result fits within the physicl memory of the mchine on which skizzo is running so the instnce is solved, or (2) n out-of-memory condition occurs. As skizzo keeps on monitoring its own resource consumption, it is ble to detect the ltter occurrence nd give up resolution-bsed resoning. The trnsition R B is triggered. As usul, the trnsition R G is followed if (nd s soon s) the current problem becomes ffordble vi SAT-bsed resoning (see the G-style). A protection mechnism is implemented ginst the unlucky possibilities tht (A) no consistent formul representtion exists when mem-out occurs, or (B) the formul yielded by SDR (or other rules) is so lrger thn the input formul tht we would prefer to restrt working on the originl version. To fce these issues, skizzo implements checkpointing mechnism: Symbolic formuls hve to be explicitly committed or rolled-bck depending on their eventul chrcteristics. This ensures tht blow-up phenomen do not negtively ffect the rest of the inference process. B: Brnching Resoning. In this sttus, serch-bsed brnching decision procedure extending the DPLL pproch to the quntified cse is pplied. Models re serched following the left-to-right prefix order of vribles during depth-first visit of the semntic evlution tree of the formul. Existentil vribles generte or nodes tht disjunctively split the brnch, universl quntifiers re ssocited to nd nodes tht split brnches conjunctively. Distinguishing fetures of skizzo: Both universl nd existentil splits re performed symboliclly. The prtil order induced by the internl structure of the quntifier tree is substituted for the left-to-right order of vribles in the prefix. The min dvntge is tht nodes with more thn one child induce sets of disjoint sub-instnces tht re solved in isoltion of one nother. After ech existentil split, the cofctored mtrix undergoes further incomplete symbolic normliztion (trnsition B S nd bck). This mechnism extends the unit-cluse-propgtion bsed form of lookhed used in purely brnching solvers. The bse cse of the recursion does not del with trivil sub-formuls. Well in dvnce, either symbolic resoning (trnsition B S, whenever the current instnce flls within its deductive power) or ground resoning (trnsition B G, whenever the ground version of the problem is ffordble) decide every sub-instnce, cting s powerful look-hed tools. Mny enhncements to the bsic DPLL procedure re implemented. A conflict-nlysis mchinery is employed in the event of inconsistent prtil ssignment to isolte the brnching steps responsible for the contrdiction to rise. This informtion is used to perform conflict-directed bckjumping. As contrdictions follow in generl from mix of brnching steps, symbolic resoning, nd SATbsed resoning, the three of these inference styles shre common conflict-nlysis engine. A symbolic lerning mechnism extrcts symbolic cluses out of contrdictions (useful to prune the rest of the serch). Size-bounded nd relevnce-bounded heuristics re used to constrint the required mount of memory. Brnching heuristics re lso enrolled: MOMS nd VSDIS re implemented. G: SAT-bsed Ground Resoning. In the G-stte we explicitly construct P rop(symbsk(f )) nd solve it vi stte-of-the-rt SAT solvers (they come out to be very efficient on such instnces). This mounts to (1) build n encoding from the structured nmespce of symbolic literls/cluses onto flt propositionl spce, (2) generte ll the necessry cluses, (3) mke the SAT solver hndle the resulting instnce: Quite some lmost-existentil fmilies of instnces re successfully delt with in the G sttus (hsh-tble bsed mechnism re implemented to mke the trnsltion fst). A trnsition x G, x {S, R, B}, is triggered s soon s the groundistion of the current formul becomes ffordble. At the beginning, this notion is simply given in terms of memory requirements: The ground version of the instnce fits into the memory nd leves enough spce for the SAT solver to work. By construction, the trnsitions x G, x {S, R} re triggered t most once, yielding n instnce SAT-equivlent to the originl QBF problem. Conversely, B genertes (possibly) long chin of SAT instnces, ech one encoding the outcome of the explortion of n entire sub-tree of the QBF semntic evlution tree. Along this chin, the notion of ffordbility is djusted by lerning lgorithm tht tries to guess the optiml switch size between B nd G. Furthermore, for the G-stte to ctively prticipte in conflict nlysis, we mp unstisfible ground cores (extrcted by nlyzing the ground inference trce) onto symbolic cores, then onto brnching choices. 6 Certifiction skizzo implements mechnism to certify its clims of (un)stisfibility. Evlution nd certifiction re completely decoupled, with lmost no overhed for the former.

7 c + () c - () e + (,b,d) e - (,b,d) f + (,b,d) f - (,b,d) b d 1 Figure 3: An exmple of st-certificte The two meshes of the chin re connected through n inference log, produced by the solver nd red by n externl certifier. The log contins informtion bout (1) the context switches between inference styles, (2) the sequence of the (symbolic) instntitions of inference rules undertken (resolutions, substitutions, ssignments), (3) entries for rollbck/commit points nd other control informtion. By reding the log forwrd, the certifier is ble to reproduce the derivtion of the empty cluse (unst instnces) nd its grph of dependencies, thus extrcting n unstisfible core. On st instnces, the certifier pplies n inductive model reconstruction[6] procedure while prsing the log bckwrd. It constructs stnd-lone, BDD-bsed st-certificte encoding QBF model. As n exmple, Figure 3 depicts the st-certificte produced for b c d e f. ( b e f) ( c f) ( d e) ( b d e) ( b c) ( c f) ( d e) ( d e) ( e f). It is possible to verify tht model is indeed encoded: By ssigning the existentil vrible e (similrly for c nd f) to TRUE when e + (, b, d) = 1 nd to FALSE when e (, b, d) = 1 the mtrix is lwys stisfied. 7 Implementtion nd experimenttion skizzo is 60k-line piece of code written in C using n object-oriented progrmming style. It hs been developed on PowerPC/McOS X pltform, then migrted onto i386/linux. It relies on the CUDD pckge, version 2.4.0, for BDD mnipultions, nd on zchff, version , for SAT solving. Among the other things, commnd-line options llow the user to individully (de)ctivte inference rules, nd to construct solving personlities by forbidding the visit of some sttes of the inference FSM. Reconstructed syntctic trees, CNF instnces nd certifictes my be dumped to secondry memory for lter nlysis. The experimentl evlution of our suite is complex tsk, involving () the ssessment of the reltive strengths of different solving personlities, (b) wide comprison with other stte-of-the-rt solvers, (c) benchmrk-centric performnce nlysis, nd (d) n nlysis of how certifiction performnces relte to solving performnces. We re crrying on ll such experiments. The reders my find updted results t [3]. Here we limit our presenttion to performnce comprison between skizzo nd b the best publiclly vilble stte-of-the-rt QBF solvers, mong which we find the three top-rted solvers ccording to most of the results presented in [20]. We consider two groups of fmilies extrcted from the QBFLIB s rchive [15]: Biere s benchmrks [8], mde up of 64 instnces divided into 4 fmilies, where the n-th instnce in ech fmily refers to model checking problem on n-bit counter, nd Ayri s benchmrks [1], mde up of 72 instnces divided into 5 fmilies, obtined from relworld verifiction problems on circuits nd protocol descriptions. None of these two benchmrks hs ever been completely solved: Their biggest instnces re quite chllenging for current solvers. The solvers we compre to re (A) QuBE-LRN [15], version 1.3, serch-bsed solver feturing lzy dt structures for unit cluse nd pure literl propgtion, plus conflict nd solution lerning; (B) Quntor [8], version , solution-bsed solver employing q-resolution nd expnsion to eliminte quntifiers, plus number of other fetures to improve efficiency; (C) SEMPROP [22], version , serch-bsed solver feturing directed bcktrcking nd lemm/model cching; (D) yquffle [31], version , serchbsed solver feturing multiple conflict-driven lerning, inversion of quntifiers nd solution-bsed bcktrcking. In Figure 4, the number of solved instnces (Y xis) is plotted ginst the (non-cumultive) logrithmic time tken to solve such instnces (X xis). skizzo is shown to outperform the other solvers on the considered benchmrks. In ddition, some of the instnces solved by skizzo hve never been solved before. 8 Conclusions The trdeoff between speciliztion nd generlity mkes the construction of hybrid lgorithms difficult tsk. Generlity (of dt structures nd mnipultion thereof) is necessry condition for different inference strtegies to cooperte. At the sme time, speciliztion is wht ech strtegy requires to work t its best. Higher-level difficulties lso exist: one inference strtegy (e.g. resolution) might trnsform instnces in so s they becomes more difficult for nother strtegy (e.g. serch). Conversely, the time tken to pply expressive forms of resoning (e.g. symbolic equivlency or hyperbinry resolution) might interfere with the tke-it-fst-ndsimple philosophy of nother policy (e.g. brnching). An idel lgorithm should be ble to tune its own inference policies. We hve obtined form of utodjustment by employing met-heuristics (wired in the inference stte-mchine), lerning (of optiml switch points mong strtegies), nd dynmic distribution (of computtionl resources ccording to n on-the-fly ssessment of rules effectiveness). Further work on the uto-djustment cpbilities of our solver is going on. References [1] A. Ayri nd D. Bsin. Bounded Model Construction for Mondic Secondorder Logics. In Proc. of CAV 00, 2000.

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