An Efficient Sequential SAT Solver With Improved Search Strategies

Transcription

1 An Effiient Sequential SAT Solver With Improved Searh Strategies F. Lu, M.K. Iyer, G. Parthasarathy, L.-C. Wang, and K.-T. Cheng K.C. Chen Department of ECE, Design Tehnology University of California at Santa Barara Cadene Corporation Astrat A sequential SAT solver Satori [1] was reently proposed as an alternative to ominational SAT in verifiation appliations. This paper desries the design of Seq-SAT an effiient sequential SAT solver with improved searh strategies over Satori. The major improvements inlude (1) a new and etter heuristi for minimizing the set of assignments to state variales, (2) a new priority-ased searh strategy and a flexile sequential searh framework whih integrates different searh strategies, and (3) a deision variale seletion heuristi more suitale for solving the sequential prolems. We present experimental results to demonstrate that our sequential SAT solver an ahieve orders-of-magnitude speedup over Satori. We plan to release the soure ode of Seq-SAT along with this paper. I. Introdution Boolean SAT finds appliations in many areas of iruit design and verifiation suh as Bounded Model Cheking [2, 16], Unounded Model Cheking [13, 14], Redundany Identifiation, Equivalene Cheking [3], Preimage Calulation [15], et. State-of-the-art SAT algorithms, as implemented in tools suh as ZCHAFF [4],BERKMIN [5], and C-SAT [6] have demonstrated that very hard SAT prolems an now e solved in reasonale time. Bounded sequential searh using SAT has een shown to e very effetive in model heking. However, its major disadvantage is its lak of ompleteness in general sequential searh. A sequential SAT solver Satori was proposed in [1], whih utilizes omined ATPG and SAT tehniques to realize a sequential SAT solver y retaining the effiieny of Boolean SAT and eing omplete in the searh. Given a sequential iruit, we assume that the iruit follows the Huffman synhronous sequential iruit model as illustrated in Figure 1. Sequential SAT (or sequential justifiation) is the prolem of finding an ordered sequene of input assignments to a sequential iruit, suh that a desired ojetive is satisfied, or proving that no suh sequene exists. A desired ojetive an e a olletion of signal value onstraints suh as that a primary output is 1. In this paper, we fous on the sequential prolems that initial states are given. PO PO PO PI PI PI PPI Cominational Logi Cominational Logi PPO PPI PPO PPI Figure 1: Bakward time-frame expansion Cominational Logi In sequential SAT, a sequential iruit is oneptually unfolded into multiple opies through akward time-frame expansion (Figure 1). In eah time-frame, the iruit eomes ominational and hene, a ominational SAT solver an e applied. In eah time-frame, a PPO state element suh as a flip-flop is translated into two orresponding signals: a pseudo primary input (PPI) and a pseudo primary output (PPO). The initial state is speified at the PPIs in time-frame 0. The ojetive is speified at the signals in time-frame n (the last timeframe, where n is unknown efore solving the prolem). While solving in an intermediate time-frames i (0< i < n 1), intermediate state solutions are produed at the PPIs and they eome intermediate ojetives for further justifiation in the previous time-frames i 1. Seq-SAT has three major improvements over Satori: State redution As pointed out in the previous work [1], minimizing the set of assignments to the state variales, alled state redution, is a ritial step for improving the effiieny of sequential searh. Seq-SAT performs state redution not only when an intermediate solution is produed ut also when an intermediate state ojetive is proved unsatisfiale. In the first ase, it employs an effiient two-step state redution algorithm to otain a smaller state lause. The algorithm follows a similar proess as the D-algorithm [8] originally proposed for ATPG and utilizes 3-value simulation. For the seond ase, a different onflit analysis proedure rather than UIP ased onflit analysis[12] is applied to derive a smaller state lause. Flexile sequential searh framework Sequential searh an e arried out in two strategies: depth-first or readth-first. Inthe depth-first strategy, a SAT algorithm expands the searh to a new timeframe whenever a solution state is identified for the urrent timeframe. If the akward justifiation fails in timeframe i 1 and no state solution an e found, it then aktraks to timeframe i and selet another state solution of timeframe i to ontinue the justifiation. In the readth-first strategy, the state solutions in the urrent timeframe i are exhausted efore expanding into a new timeframe i 1. In this paper, we provide a flexile sequential searh framework in whih these two searh strategies an e integrated. We introdue a priority-ased searh strategy that an omine oth depthfirst searh and readth-first searh and demonstrate its effiieny over other searh strategies. Deision variale seletion heuristi InSAT,theideathat make deisions to satisfy reently dedued lauses[4, 5] has een proven to e very effetive for solving hard ominational prolems. However, when applying a ominational solver for sequential SAT, the run time of ominational SAT solving is not neessarily the most important fator to e optimized. In this paper, we propose a deision variale seletion heuristi whih is more suitale for solving sequential prolems. The rest of the paper is organized as the following. Setion II illustrates the use of ominational SAT in sequential SAT. In Setion III, we disuss the importane of state redution and desrie our approahes to this prolem. Setion IV presents our sequential searh framework. In Setion V, we present our priority-ased searh strategy. Setion VI disusses the impat of deision ordering in ominational SAT on the overall sequential SAT effiieny. Setion VII summarizes experimental results to demonstrate the superiority of our urrent sequential SAT solver. Setion VIII onludes the paper.

2 II. Cominational and sequential SAT Given a finite set of variales, V, over the set of Boolean values B {0,1}, aliteral, l/l is an instane of a variale or its omplement, v/ v V. A lause i, is a disjuntion of literals (l 1 l 2... l n ). Aformula f, is a onjuntion of lauses n. A lause is onsidered as a set of literals, and a formula as a set of lauses. An assignment A satisfies a formula f if f (A)=1. An assignment is alled maximal when every variale in V reeives a value assignment. Given a sequential iruit, a state lause is a lause onsisting only of state variales. In a Boolean Satisfiaility (SAT) prolem (or here all it ominational SAT to differentiate from sequential SAT), a formula f is given and the prolem is to find an assignment A to satisfy f or prove that no suh an assignment exists. SAT has attrated tremendous researh effort in reent years, resulting in the developments of various effiient SAT solver pakages suh as [4, 5, 6, 10, 11]. Through akward timeframe expansion (Figure 1), a sequential SAT prolem an e translated into a sequene of ominational SAT prolems. II-A. Sequential SAT and state lauses Initial state: (x=1, y=0, z=1) (1) time frame n time frame n a 1 a (2) f=1 0 0 f=1 PPI x PPI y PPI z PPO x 0 PPI x 1 PPI y 0 PPI z PPO x time frame (n 1) state time frame n a ojetive a f=1 PPI x PPO x 0 PPI x PPO x (3) PPI y PPI z 1 PPI y PPI z state lause: (x+y ) state lause: (x+y )(x) (4) No solution: UNSAT Figure 2: Sequential SAT with state lauses To illustrate the usage of state lauses in sequential SAT, Figure 2 depits a simple example iruit with three primary inputs a,,, one primary output f, and three state elements x,y,z. The initial state is (x = 1,y = 0,z = 1). Suppose the SAT ojetive is to satisfy f = 1. Starting from time-frame n where n is unknown, the iruit is translated into a ominational opy with state variales dupliated as PPOs and PPIs. This is illustrated as (1) in the Figure. Sine this represents a ominational SAT prolem, a Boolean SAT solver an e applied. Suppose after the ominational SAT solving, a solution (a = 1, = 1, = 0,PPI x = 0,PPI y = 1,PPI z = 0) for satisfying f = 1 (step (2))is identified. Notie that all inputs have values assigned in this solution, even though it might not e neessary to have all of them assigned in order to satisfy f = 1. This phenomenon is due to the use of a ominational SAT solver as mentioned earlier. The assignments at PPIs implies a solution state (x = 0,y = 1,z = 0) that doesn t ontain the initial state. Instead of justifying the state (x = 0,y = 1,z = 0) in time-frame n 1, we an examine it first to hek whether we an hange as many of the assignments to don t-ares while still satisfying the ojetive f = 1. Suppose after analysis, we determine that z = 0 is unneessary, it an e removed from the solution state (x = 0,y = 1,z = 0) and a solution state (x = 0,y = 1) is derived. We all this step solution state redution sine it tries to remove the unneessary state variale assignments from a solution. The solution state (x = 0,y = 1) doesn t ontain the initial state either, so it still needs to e justified in timeframe n 1. Bakward time-frame expansion an e ahieved y adding a state ojetive (PPO x = 0, = 1) to the ominational opy of the iruit. Also, a state lause (x+y ) is generated to prevent reahing state solutions ontained y the state su-spae (x = 0,y = 1) in time-frame n 1. The new ominational SAT instane is then passed to the Boolean SAT solver. Suppose in time-frame n 1, no solution an e found for state ojetive (PPO x = 0, = 1). Then, we aktrak to time-frame n to find another solution. In a way, we have proved that from state (x = 0,y = 1), there exists no solution. Therefore, there is no need to remove the state lause (x + y ). However, at this point it is enefiial to perform further analysis to determine if oth PPO x = 0and = 1 are involved in the onflit that indiates state ojetive (PPO x = 0, = 1) is unsatisfiale. Suppose after onflit analysis, It is disovered that only PPO x = 0 is involved in the onflit. A state lause (x) is then added. The future ominational SAT instanes will then have the state lauses (x + y )(x) inluded, that reord the non-solution state su-spaes previously identified. This is illustrated in (4) of the Figure. We all this step ojetive state redution sine it tries to derive a smaller state lause when a state ojetive eomes unsatisfiale. The solving ontinues until either one of the following two onditions is reahed: 1. After solution state redution, a solution is found whose state part ontains the initial state. For example, a solution with the state part (x = 1,z = 1) ontains the initial state (x = 1,y = 0,z = 1). In this ase, a solution for the sequential SAT prolem is found. Note that a solution without any assignments to PPIs ontains any initial state. 2. If aktraked to time-frame n, the initial ojetive f = 1 annot e satisfied under the onstraints imposed y all the state lauses added, then the original prolem is unsatisfiale. This is equivalent to say: if any ojetive (inluding the initial ojetive f = 1 and all intermediate ojetives) annot e satisfied under the onstraints imposed y all the state lauses added, then the original prolem is unsatisfiale. The aove example illustrates several important onepts in the design of our urrent sequential SAT solver. The state redution involves finding smaller state lauses in order to more effetively prune the searh spae. There are two types of state redution. One is solution state redution, the other is ojetive state redution. The use of state lauses serves two purposes: (1) to reord those state su-spaes that have een explored, and (2) to reord those state su-spaes ontaining no solution. The first is to prevent the searh from entering a state justifiation loop. Although oneptually the searh follows the akward timeframe expansion, the aove example demonstrates that in the implementation, expliit time-frame expansion is not neessary. In other words, a sequential SAT solver needs only one opy of the iruit. Moreover, there is no need to memorize the numer of time-frames eing expanded. Later we will disuss our implementation of the solver and demonstrate how this an e ahieved y using an ojetive list. The aove example demonstrates the use of depth-first searh strategy. However, y using state lauses, more flexile searh strategies are feasile if the intermediate state ojetives are reorded in a list. The idea of the sequential searh framework an e desried as follows: Given a sequential prolem with an ojetive oj and an initial state s 0, C is the one time-frame ominational opy of the sequential iruit where eah state variale is expanded to a PPI variale and a PPO variale. Suppose we use an ojetive list FO to store the intermediate ojetives, and initially it only ontains oj. Eah time, an ojetive o is seleted from FO and given to the ominational iruit solver.

3 The iruit solver then solves o on C. There will e three possile outomes reported y the solver: (1) o is unsatisfiale, (2) at least one state in the otained solutions ontains the initial state s 0, and (3) no state in the otained solutions ontains the initial state s 0. For ase (1), we an remove o from FO. For ase (2), the searh stops and the sequential prolem is proven satisfiale. For ase (3), the state part of eah solution is added to FO as a new state ojetive and a state lause is added to C to prevent the state e explored and found again in any future solutions. If finally, the ojetive list FO eomes empty, whih means all ojetives have een proven to e unsatisfiale, then the searh stops and the sequential prolem is proven unsatisfiale. In this framework, the keys are to deide how to selet the next ojetive from the list for solving and to deide how many solutions to e found for an ojetive eah time. The details will e giveninsetioniv. In the following, we will desrie the detail of Seq-SAT. To failitate the desription, we define the following two terms: (1) a frame ojetive is an ojetive to e satisfied whih is passed to the ominational SAT solver. A frame ojetive an e either the initial ojetive or a state ojetive. (2) a frame solution is an assignment at the PIs and PPIs, whih satisfies a given frame ojetive. III. State redution for state spae pruning There are two types of state redution: solution state redution (SSR) and ojetive state redution (OSR). SSR is performed when a frame solution is identified y the ominational SAT solver. OSR is applied when a state ojetive eomes unsatisfiale. The details are disussed in the following. III-A. Solution state redution When a frame solution is identified y the ominational SAT solver, all PIs and PPIs have assigned values. Our goal is to find and un-assign the unneessary assignments at the PPIs. With fewer assigned PPIs, a smaller state lause an e reated, whih prunes a larger state su-spae. Given an initial frame solution, to derive one ontaining a minimal numer of state variales with value assignment is an NP-omplete prolem. Therefore, we only derive a loally minimized 3-value solution. A frame solution s of a frame ojetive o is a loally minimized 3-value solution if and only if un-assign any state variale of s (hanging its value from a 0/1 to X), the resulting assignments annot satisfy o y 3-value simulation. One straight forward heuristi to derive a loally minimized 3- value solution is to employ 3-value simulation. That is, for all assigned state variales of a solution, un-assign one of them at a time to hek if its assignment is neessary to satisfy the frame ojetive. If not, the assignment is removed from the solution, this proedure ontinues until eah assigned state variale has een heked one, then the remaining assignments form a 3-value minimized solution. A frame solution reported y a SAT solver usually ontains many unneessary assignments at state variales. Diretly apply 3-value simulation ould e omputationally expensive. For example, removing N unneessary state assignments needs at least N times of simulation even if parallel simulation is used. To redue the numer of simulation runs and improve effiieny, we adopt a two-step state redution approah. In this approah, a modified D-algorithm [8] is first applied to otain a minimized solution. Given a frame ojetive O and its frame solution S, the modified D-algorithm justify O ased on S, that is, when the D-algorithm makes a deision at a signal, the value assigned to the signal has to e onsistent with its value in the solution S. Hene, the D-algorithm is used as a trae proedure, not as a searh proedure. Based on the minimized solution from the first step, 3-value simulation is then employed to derive a 3-value minimized solution. This approah is ased on the assumption that the D-algorithm an usually find a solution ontaining muh fewer assignments than that of a SAT solver. Experiments show our assumption is valid in general. The details of the approah are omitted due to page limit. TABLE I: RESULTS OF SOLUTION STATE REDUCTION Ciruit Org # Step 1 resulting Step 2 resulting # %redued times(se) # %redued times(se) s % % 1 s % % 1 s % % 2 s % % 3 s % % 100 s % % 110 Tale I shows experimental results to demonstrate the impat of solution state redution. All our experiments were run on P4 2GHz Linux mahines with 1 GB memory. In these experiments, a sequene of sequential SAT prolem instanes are reated for eah iruit. For eah iruit and eah of its primary outputs f, the first SAT ojetive is to satisfy f = 1 and then we would try to satisfy f = 0. This is a typial primary output toggle experiment. The results shown for eah iruit are for solving all toggle SAT instanes for the iruit. We assume the initial state is that all state elements have a 0 value. Note, this kind of prolems may ontain very diffiult ases even for ISCAS 89 iruits. The two # olumns show the numers after state redution with modified D-algorithm (step 1) and then, with three value simulation (step 2), respetively. The two % redued olumns show the orresponding perentage of redution. The two run-time olumns show the run times ased on using step 1 and then, using step 2. We note that if no state redution is performed, then the sequential SAT might not finish solving all instanes within a reasonale time for eah of these examples. From the results, we an see that the modified D-algorithm ould redue most of the unneessary assignments to the state variales, so the numer of 3-value simulation runs an e greatly redued. Although step 2 does not help muh in terms of the total perentage of assignment redution, it does help further improve the overall performane. III-B. Ojetive state redution As a state so is added to the ojetive list, a state lause s is added to prevent the state from eing reahed again. When the state ojetive so eomes unsatisfiale, instead of using the same state lause s, a smaller state lause might e added to prune a larger searh spae. This an e ahieved y additional onflit analysis when a state ojetive is proved unsatisfiale. We explain the idea y an example, suppose the state ojetive so is (PPO r = 1,PPO s = 1,PPO t = 0,PPO x = 0, = 1), the impliation graph when s is proved unsatisfiale (whih means there is a onflit that an not e resolved y aktrak) is shown as in Figure 3. From Figure 3, if traing from the onflit points to the PPO state variales, we an dedue that (PPO r = 1,PPO s = 1,PPO t = 0) is unsatisfiale, so the state lause (r + s + t), whih is smaller than (r + s + t + x + y ), an e added to the iruit. However, we need to e areful that the state lauses added in this way shouldn t exlude the searh spae that ontains the initial state. Otherwise a satisfiale prolem erroneously eomes unsatisfiale sine the initial state ouldn t e reahed anymore after adding suh a state lause. Using the aove example, if the values of r, s and t in the initial state are 1, 1 and 0 respetively, state lause (r + s +t) would exlude the initial state. To avoid this, we an just selet a state variale from the state ojetive whose ojetive value is not its initial value and add its orresponding literal to the state lause. For example, if the value of x in the initial state is

4 1 in the aove example, we an add x to the state lause (r + s +t) to form state lause (r + s + t + x ) whih an e added to the iruit. This an always e done sine state ojetives don t ontain the initial state. That is, eah state ojetive must have at least one variale whose ojetive value is not same as its value in the initial state, otherwise, the prolem should has een proved satisfiale. This redution is valid even if state lauses that were used for preventing justifiation loop are involved in the onflit graph. We also note that this redution is not related to identify the suset of a CNF formula that is suffiient for unsatisfiaility, instead, it just tries to find a onflit lause that ontains only literals of the variales in a state ojetive. Tale II gives the experimental results omparing the performane of our sequential solver with and without OSR. These are the same toggle experiments as those in Tale I. TABLE II: EFFECTS OF OBJECTIVE STATE REDUCTION Without OSR With OSR Ciruit runtimes(se) stats runtimes(se) stats s /0/0 1 12/0/0 s /0/0 1 10/0/0 s /5/4 3 90/8/0 s /58/ /63/13 s /54/ /69/5 s /0/ /0/0 s /0/ /0/0 s /58/ /58/0 **aort time for eah prolem is 100 se. stats: x/y/z are #SAT/#UNSAT/#Aort From the experiment results, it an e seen that the performane is greatly improved for half of the testases y OSR, and only for one testase, it auses a small overhead. PPO PPO t PPO x PPO s y PPO r a e d f f Confliting Variale Figure 3: An impliation graph when a state ojetive is proved unsatisfi ale IV. Flexile sequential searh framework In our sequential SAT design, state lauses are aumulated through the solving proess. The sequential solving proess onsists of a sequene of ominational solving tasks ased on a given frame ojetive. At the eginning, the frame ojetive is the initial ojetive. As the solving proeeds, many frame solutions eome frame ojetives. These frame ojetives are stored in the ojetive list. The use of state lauses and the ojetive list offer muh flexiility in the hoie of the searh strategy. Different searh strategies an e implemented y using different riteria in seleting the next frame ojetive for solving, and in deiding how many solutions to e otained at a time for a frame ojetive. For example, for the depth-first searh, the ojetive list is used as a stak where the next seleted ojetive is the one most reently produed. For the readth-first searh, the ojetive list is used as a queue. In addition, in the readth-first searh, it demands the solver to find all solutions for the seleted ojetive. After that, the ojetive is removed from the ojetive list. This framework further enales us to explore various searh strategies etween these two extreme ones whih will e desried later in this setion. The overall framework of our sequential SAT solver is desried in Algorithm IV.1. In the desription, we assume that eah frame ojetive produes at most one frame solution at a time. This assumption is only for purpose of easier explanation. The algorithm an e easily modified to serve the ase that multiple solutions are produed at a time. The proedure PPO state onflit analysis() does ojetive state redution, and state redution() does solution state redution. The proedure selet a frame ojetive() selets a frame ojetive to e solved next, upon different implementation, various searh strategies an e realized, and the details will e given in the next setion. A frame ojetive an e removed from the ojetive list only if it is proven unsatisfiale y the ominational SAT. If it is satisfiale, the frame ojetive stays in the ojetive list. If the ojetive list eomes empty during the sequential searh, it means that the solver has exhausted all the state ojetives and, therefore, proved that the initial ojetive is unsatisfiale. Note that during eah step of ominational SAT, onflit lauses are also aumulated through the ominational SAT solving proess. When the sequential solving swithes from one frame ojetive to another, these onflit lauses stay. Hene, in the sequential solving proess, the onflit lauses generated y the ominational SAT are also aumulated. Our experiene indiates that although these onflit lauses does help speed up the ominational SAT solving, for sequential SAT, the state lauses are the dominating soure for the effiieny improvement of the overall sequential searh. Algorithm IV.1: SEQUENTIALSOLVER(C,o j,s 0 ) omment: C is the iruit with PPIs and PPOs expanded omment: o j is the initial ojetive omment: s 0 is the initial state omment: FO is the ojetive list FO {o j}; while (FO /0) f o j selet a frame ojetive(fo); f sol ominational solve a frame ojetive(c, f o j ); if ( f sol = NULL) lause PPO state onflit analysis(c, f o j ); add state lause(c,lause); FO FO {f o j }; do state ue state redution(c, f o j, f sol ); if { (s 0 state ue) return else (SAT); lause onvert to lause(state ue); else add state lause(c,lause); FO FO+ {state ue}; return (UNSAT) V. On the sheduling of frame ojetives In this setion, we introdue a new searh strategy, alled the priority-ased searh strategy, whih an e easily implemented in the searh framework desried aove. We further ompare it with the depth-first and readth-first searh strategies and demonstrate its superiority. Given a state ojetive, we define its size as the total numer of state variales minus the numer of state variales with an inary ojetive value (i.e. not an unknown X) in the state ojetive. We use size(o j) to denote the size of o j. Note that a state ojetive with alargersize orresponds a larger state su-spae. Hene, if there are multiple state ojetives in the ojetive list, it seems intuitive

5 to hoose the largest state ojetive as the next ojetive for solving as it orresponds to exploring the solution with the largest state su-spae, and its solutions, if any, tend to ontain fewer state assignments, whih means smaller state lauses are likely to e generated to prune a larger searh spae. We further define the ount of an ojetive o j as the numer of times that o j has een seleted for solving y the ominational solver. If we use ount to guide the seletion of ojetives and hoose the one with the largest ount as the next ojetive, it will e more like the readth-first searh strategy. On the other hands, if we hoose the one with the smallest ount, It will e more like the depth-first searh. Now we define priority of a frame ojetive o j, pri(o j), as follows: pri(o j) =size(o j) ount(o j)/10 (1) In our implementation of the priority-ased searh strategy, we selet the ojetive with the largest value of pri. If there is a tie under this seletion riterion, then the most reently generated one will e seleted. For eah seleted ojetive, at most one solution is otained at a time. Through extensive experiments, we oserve that omining size and ount in this way maximizing the performane, in general, for oth satisfiale and unsatisfiale ases. With the ojetive list, depth-first and readth-first searh strategies an e implemented easily. In depth-first searh, the list ehaves like a stak. Hene, the seleted ojetive is the one most reently produed. In readth-first searh, the list ehaves like a queue. However, in readth-first searh, solving an ojetive means to finish produing all possile solutions. Then, the ojetive is removed from the ojetive list. Tale III gives the experimental results omparing different searh strategies. These are the same toggle experiments as those in Tale I. The priority olumn is the same as the last times olumn in Tale I. As it an e seen, the new searh strategy signifiantly outperforms the depth-first and the readth-first strategies. TABLE III: CPU RUNTIMES (IN SECONDS) ON SEARCH STRATEGIES depth-fi rst readth-fi rst priority Ciruit runtimes stats runtimes stats runtimes stats (se) (se) (se) s /0/0 2 12/0/0 1 12/0/0 s /0/ /0/1 1 10/0/0 s /8/ /5/4 3 90/8/0 s /54/ /63/ /63/13 s /60/ /59/ /69/5 s /0/ /0/ /0/0 s /0/ /0/ /0/0 s /57/ /57/ /58/0 **aort time for eah prolem is 100 se. stats: x/y/z are #SAT/#UNSAT/#Aort VI. On the deision variale seletion heuristis in the ominational SAT solver For pure ominational SAT, oth VISDS[4] deision variale seletion and lause ased deision variale seletion [5] have een proven effetive. However, when applying a ominational SAT solver for sequential SAT, the runtime of ominational SAT solving is not the only ritial fator. In eah run of ominational SAT solving during the sequential SAT, the smaller the neessary assignments are made at the PPIs, the more effiiently the state spae an e pruned. Therefore, the deision variale seletion heuristis in ominational SAT also need modifiations. To demonstrate this point, we onduted experiments to ompare different deision variale seletion heuristis in ominational SAT. Deision heuristi 1: This is the same as the VISDS deision variale seletion [4]. Deision heuristi 2: First, the deision variale is seleted from the most reently generated onflit lause that is not yet satisfied [5]. If all the generated onflit lauses are satisfied, the deision variale is seleted ased on the J-nodes [17, 6] with the highest topologial order (losest to the POs and the PPOs). This deision heuristi is used in the original ominational SAT solver C-SAT [6]. Deision heuristi 3: First, the deision variale is seleted from the most reently generated onflit lause that is not yet satisfied as in Heuristi 2 aove. If all the generated onflit lauses are satisfied, the deision variale is seleted from PIs and PPIs where PIs are seleted efore PPIs This is the default heuristi used in our final implementation. The reasons we adopt Heuristi 3 in Seq-SAT are summarized as follows: (1). Using input variales as deision points results in etter performane for easy ominational prolems in general. For a sequential prolem that an e transformed into a series of relatively easy ominational prolems, this heuristi works very well. (2). For hard prolems, as the onflit lauses tend to aumulate fast during searh, the deisions will gradually e dominated y the onflit lauses. Therefore, the performane will not degrade muh. (3). Seleting PIs prior to PPIs as deision variales tends to result in solutions with fewer state variales with value assignments [14]. TABLE IV: CPU RUNTIMES ON DIFFERENT DECISION HEURISTICS heuristi 1 heuristi 2 heuristi 3 Ciruit runtimes stats runtimes stats runtimes stats (se) (se) (se) s /0/0 3 12/0/0 1 12/0/0 s /0/2 1 10/0/0 1 10/0/0 s /7/ /7/1 3 90/8/0 s /62/ /63/ /63/13 s /69/ /69/ /69/5 s /0/ /0/ /0/0 s /0/ /0/ /0/0 s /58/ /58/ /58/ /1/0 13 0/1/0 20 0/1/ /1/0 16 0/1/0 8 0/1/0 **aort time for eah prolem is 100 se. stats: x/y/z are #SAT/#UNSAT/#Aort Tale IV shows the results of different heuristis. The experiments for s- iruits were the same PO toggle experiments iruit as explained efore. The experiments for - iruits were the ominational equivalene heking prolem using the miter iruit of two idential ISCAS 85 iruits. It an e seen that the deision heuristi 3 works well for sequential iruits, even for large ominational iruits, its performane is not ad. VII. Additional experimental results Tale V ompares our solver with NuSMV [16] ounded model heking funtion for satisfiale ases. We have shown in the previous experiments how eah idea works, Tale VI ompares the overall performane of Seq-SAT with Satori. In Tale V, the Property olumn gives the LTL formula for eah testase, and length olumns give the length of the witness vetors otained y Seq-SAT and NuSMV respetively. For NuSMV, the version we used is NuSMV zhaff in whih the SAT solver zhaff is used for ounded model heking. The initial states used in these examples are all 0.

6 TABLE V: COMPARISON TO NUSMV BOUNDED MODEL CHECKING Seq-SAT NuSMV Ciruit Property runtimes(se) length runtimes(se) length s526 G(g214 = 0) s13207 G(g594 = 0) s13207 G(g785 = 0) s38417 G(g5549 = 0) > s38417 G(g16399 = 0) s38584 G(g11678 = 0) s38584 G(g29212 = 0) > **aort time for eah prolem is 3600s. From Tale V, it an e seen that, even though the witness vetors derived y Seq-SAT are in general longer than those y NuSMV, Seq- SAT learly outperforms NuSMV in terms of CPU runtime. In Satori, the depth-first searh and the readth-first searh are implemented separately. So in Tale VI, we ompare Seq-SAT with Satori under these two searh strategies for the same toggle experiments as those in Tale I. Tale VII summarizes the numer of the aort ases shown in Tale VI. Column #Satori aort shows the numer of ases aorted in oth Satori depth-first searh and Satori readth-first searh. Column #SSANA gives the numer of ases aorted in Seq-SAT ut not aorted in either Satori depth-first searh or Satori readth-first searh. TABLE VI: COMPARISON TO SATORI Satori depth-fi rst Satori readth-fi rst Seq-SAT Ciruit runtimes stats runtimes stats runtimes stats (se) (se) (se) s /0/4 5 12/0/0 1 12/0/0 s /0/ /0/1 1 10/0/0 s /4/ /4/ /8/0 s /48/ /59/ /63/13 s /52/ /52/ /69/5 s /0/ /0/ /0/0 s /0/ /0/ /0/0 s /57/ /58/ /58/0 **aort time for eah prolem is 100 se. stats: x/y/z are #SAT/#UNSAT/#Aort TABLE VII: ABORTED CASES ANALYSIS Ciruit #Satori aort #Seq-SAT aort #SSANA s s s s s s s s It an e seen that Seq-SAT ahieves signifiant performane improvement for almost all ases. It is also interesting to note that the results of the depth-first searh in Satori are quite different from those of depth-first searh implemented in Seq-SAT (see Setion V Tale III). This indiates that the depth-first searh is quite unstale (espeially for satisfiale ases) and easy falls into ad searh area. VIII. Conlusion In this paper, we present an effiient sequential SAT solver Seq- SAT. The implementation of Seq-SAT employs four new ideas: (1) etter state redution algorithms, (2) a more flexile searh framework for aommodating and integrating different searh strategies, (3) the priority-ased searh strategy, and (4) a modified deision variale seletion heuristi for the underlying ominational iruit SAT solver. With these new ideas and effiient data strutures and implementation, Seq-SAT ahieves very signifiant speedup over Satori. We will release the soure ode of Seq-SAT along with the puliation of this paper. Referenes [1] M. K. Iyer, G Parthasarathy, K.-T. Cheng, SATORI - A Fast Sequential SAT Engine for Ciruits. Pro. IEEE/ACM International Conferene on Computer-Aided Design [2] A. Biere, A. Cimatti, E. M. Clarke, M. Fujita, and Y. Zhu, Symoli Model Cheking using SAT Proedures instead of BDDs. In Pro. of the 36th ACM/IEEE Design Automation Conf., pp , June [3] P. Tafertshofer, A. Ganz, and K. Antreih, IGRAINE An Impliation-Graph ased Engine for Fast Impliation, Justifiation and Propagation. IEEE Trans. on Computer-Aided Design, 19(8): , August [4] M. Moskewiz, C. Madigan, Y. Zhao, L. Zhang, and S. Malik, Chaff: Engineering an effiient SAT solver. Pro. of ACM/IEEE Design Automation Conf., pp , June [5] E. Golderg and Y. Novikov, BerkMin: A fast and roust Sat Solver. Pro. European Design and Test Conf., pp , Marh [6] Feng Lu, Li-C. Wang, Kwang-Ting Cheng, A iruit SAT solver with signal orrelation guided learning. In Pro. European Design and Test Conferene [7] P. Tafertshofer, A. Ganz, and M. Henftling, A SAT-Based Impliation Engine for Effiient ATPG, Equivalene Cheking, and Optimization of Netlists. Pro. IEEE/ACM International Conferene on Computer-Aided Design, pp , Nov [8] M. Aramovii, M. A. Breuer, and A. D. Friedman. Digital Systems Testing and Testale Design. CS Press, 1 st ed., [9] G Parthasarathy, M. K. Iyer, Li-C. Wang and K.-T. Cheng, Effiient Reahaility Cheking and Pre-image Computation using Sequential SAT. Pro. Asia and South Paifi Design Automation Conf [10] H. Zhang. SATO: An Effiient Propositional Prover. Pro. of International Conferene on Automated Dedution, Vol 1249, LNAI, pp , 1997 [11] J. P. Marques-Silva and K. A. Sakallah. GRASP: A Searh Algorithm for Propositional Satisfiaility. IEEE Trans on Computers, vol.48, pp , [12] L. Zhang, C. Madigan, M. Moskewiz, and S. Malik. Effiient onflit driven learning in a Boolean satisfiaility solver. Pro. IEEE/ACM International Conferene on Computer-Aided Design, pp , Nov [13] K. L. MMillan. Applying SAT methods in unounded symoli model heking. Pro. of International Conferene on Computer Aided Verifiation, Vol 2404, LNSC, pp , [14] H.-J. Kang, I.-C. Park. SAT-Based Unounded Symoli Model Cheking. Pro. of ACM/IEEE Design Automation Conf., pp , June [15] B. Li, M. S. Hsiao, and S. Sheng. A novel SAT all-solutions solver for effiient preimage omputation. Pro. of Design, Automation and Test in Europe Conf., pp , Fe [16] A. Cimatti, E. Clarke, E. Giunhiglia, et. al.. NuSMV 2: An OpenSoure Tool for Symoli Model Cheking. In Pro. CAV 02, LNCS. Springer Verlag, [17] M.K.Ganai, L.Zhang, P.Ashar, A.Gupta, and S.Malik. Comining strengths of iruit-ased and CNF-ased algorithms for a high-performane SAT solver. In Pro. ACM/IEEE Design Automation Conf., pp , June 2002.