A Compressed Breadth-First Search for Satisfiability

Transcription

1 A Compressed Breadth-First Searh for Satisfiaility DoRon B. Motter and Igor L. Markov Department of EECS, University of Mihigan, 1301 Beal Ave, Ann Aror, MI dmotter, Astrat. Leading algorithms for Boolean satisfiaility (SAT) are ased on either a depth-first tree traversal of the searh spae (the DLL proedure [6]) or resolution (the DP proedure [7]). In this work we introdue a variant of Breadth- First Searh (BFS) ased on the aility of Zero-Suppressed Binary Deision Diagrams (ZDDs) to ompatly represent sparse or strutured olletions of susets. While a BFS may require an exponential amount of memory, our new algorithm performs BFS diretly with an impliit representation and ahieves unonventional redutions in the searh spae. We empirially evaluate our implementation on lassial SAT instanes diffiult for DLL/DP solvers. Our main result is the empirial Θ n 4 µ runtime for hole-n instanes, on whih DLL solvers require exponential time. 1 Introdution Effiient methods to solve SAT instanes have widespread appliations in pratie and have een the fous of muh reent researh [16, 17]. Even with the many advanes, a numer of pratial and onstruted instanes remain diffiult to solve. This is primarily due to the size of solution spaes to e searhed. Independently from SAT, [9] explored Lempel-Ziv ompression in exhaustive searh appliations suh as game-playing and ahieved memory redutions y a small onstant fator. Text searhes in properly indexed dataases ompressed using a Burrows- Wheeler sheme were proposed in [8]. However, these works enale only a limited set of operations on ompressed data, and the asymptoti ompression ratios are too small to hange the diffiulty of searh for satisfiaility. A different type of ompression is demonstrated y Redued Ordered Binary Deision Diagrams (ROBDDs) and Zero-Suppressed Binary Deision Diagrams (ZDDs) [13]. In general, Binary Deision Diagrams impliitly represent ominatorial ojets y the set of paths in a direted ayli graph. Complexity of algorithms that operate on BDDs is often polynomial in the size of the BDD. Therefore, y reduing the size of BDDs, one inreases the effiieny of algorithms. There have een several efforts to leverage the power of BDDs and ZDDs as a mehanism for improving the effiieny of SAT solvers. [4] [5] used ZDDs to store a lause dataase and perform DP on impliit representations. [2] used ZDDs to ompress the ontainers used in DLL. Another method is to iteratively onstrut the BDD orresponding to a given CNF formula [19]. In general this method has a different lass of tratale instanes than DP [11]. Creating this BDD is known to require exponential time for several instanes [11], [19]. For random 3-SAT instanes, this method also gives slightly different ehavior than DP/DLL solvers [12].

2 The goal of our work is to use ZDDs in an entirely new ontext, namely to make Breadth-First Searh (BFS) pratial. The primary disadvantage of BFS exponential memory requirement often arises as a onsequene of expliit state representations (queues, priority queues). As demonstrated y our new algorithm Cassatt, integrating BFS with a ompressed data struture signifiantly extends its power. On lassial, hard SAT enhmarks, our implementation ahieves asymptoti speed-ups over pulished DLL solvers. To the est of our knowledge, this is the first work in pulished literature to propose a ompressed BFS as a method for solving the SAT deision prolem. While Cassatt does not return a satisfying solution if it exists, any suh SAT orale an e used to find satisfying solutions with at most Vars alls to this orale. The remaining part of the paper is organized as follows. Setion 2 overs the akground neessary to desrie the Cassatt algorithm. A motivating example for our work is shown in Setion 3. In Setion 4 we disuss the impliit representation used in Cassatt, and the algorithm itself is desried in Setion 5. Setion 6 presents our experimental results. Conlusions and diretions of our ongoing work are desried in Setion 7. 2 Bakground A partial truth assignment to a set of Boolean variales V is a mapping t : V 0 1. For some variale v ¾ V, if t vµ 1 then the literal v is said to e true while the literal v is said to false (and vie versa if t vµ 0). If t vµ, v and v are said not to e assigned values. Let a lause denote a set of literals. A lause is satisfied y a truth assignment t iff at least one of its literals is true under t. A lause is said to e violated y a truth assignment t if all of its literals are false under t. A Boolean formula in onjuntive normal form (CNF) an e represented y a set C of lauses. 2.1 Clause Partitions For a given Boolean formula in CNF, a partial truth assignment t is said to e invalid if it violates any lauses. Otherwise, it is valid. The impliit state representation used in the Cassatt algorithm relies on the partition of lauses implied y a given valid partial truth assignment. Eah lause must fall into exatly one of the three ategories shown in Figure 1. We are going to ompatly represent multiple partial truth assignments that share the same set of assigned variales. Suh partial truth assignments appear in the proess of performing a BFS for satisfiaility to a given depth. Note that simply knowing whih variales have een assigned is enough to determine whih lauses are unassigned or not ativated. The atual truth values assigned differentiate satisfied lauses from open lauses. Therefore, if the assigned variales are known, we will store the set of open lauses rather than the atual partial truth assignment. While the latter may e impossile to reover, we will show in the following setions that the set of open lauses ontains enough information to perform a BFS. Another important oservation is that only lauses whih are ut y the vertial line in Figure 1 have the potential to e open lauses. The numer of suh ut lauses

3 Unassigned Clauses: Clauses whose literals are unassigned. Satisfied Clauses: Clauses whih have at least one literal satisfied. Open Clauses: Clauses whih have at least one, ut not all of their literals assigned, and are not satisfied. a (e +f) (++d) (h + j) (a+) (g + d e f g h i i+j) j Fig. 1. The three lause partitions entailed y a valid partial truth assignment. The vertial line separates assigned variales (a through e) from unassigned variales ( f through j, grayed out). We distinguish (i) lauses to the right of the vertial line, (ii) lauses to the left of the vertial line, and (iii) lauses ut y the vertial line. does not depend on the atual assigned values, ut depends on the order in whih variales are proessed. This oservation leads to the use of MINCE [1] as a heuristi for variale ordering sine it redues utwidth. MINCE was originally proposed as a variale ordering heuristi to omplement nearly any SAT solver or BDD-ased appliation. MINCE treats a given CNF as a (undireted) hypergraph and applies reursive alaned min-ut hypergraph isetion in order to produe an ordering of hypergraph verties. This algorithm is empirially known to ahieve small values of total hyperedge span. Intuitively, this algorithm tries to minimize utwidth in many plaes, and as suh is an ideal omplement to the Cassatt algorithm. Its runtime is estimated to e O V Cµlog 2 Vµ, where V is the numer of variales and C is the numer of lauses in a given CNF. By reduing the utwidth in Cassatt, we otain an exponential redution in the numer of possile sets of open lauses. However, storing eah set expliitly is still often intratale. To effiiently store this olletion of sets, we use ZDDs. 2.2 Zero-Suppressed Binary Deision Diagrams In Cassatt we use ZDDs to attempt to represent a olletion of N ojets often in fewer than N its. In the est ase, N ojets are stored in O p lognµµ spae where p nµ is some polynomial. The ompression in a BDD or ZDD omes from the fat that ojets are represented y paths in a direted ayli graph (DAG), whih may have exponentially more paths than verties or edges. Set operations are performed as graph traversals over the DAG. Zero-Suppressed Binary Deision Diagrams (ZDDs) are a variant of BDDs whih are suited to storing olletions of sets. An exellent tutorial on ZDDs is availale at [14]. A ZDD is defined as a direted ayli graph (DAG) where eah node has a unique lael, an integer index, and two outgoing edges whih onnet to what we will all T- Child and E-hild. Beause of this we an represent eah node X as a 3-tuple X n X T X E where n is the index of the node X, X T is its T-Child, and X E is its E-Child. Eah path in the DAG ends in one of two speial nodes, the 0 node and the 1 node. These nodes have no suessors. In addition, there is a single root node. When we use a ZDD we will in reality keep a referene to the root node. The semantis of a ZDD an e defined reursively y defining the semantis of a given node.

4 a BFS Queue: a BFS Queue: a BFS Queue: a {0, 1} {01, 10, 11} {011, 100, 101} d d d BFS Queue: {0110, 0111, 1000, , 1011} 1 Step 1 Step 2 Step 3 Step 4 Fig. 2. Steps Taken y Breadth-First Searh A ZDD an e used to enode a olletion of sets y enoding its harateristi funtion. We an evaluate a funtion represented y a ZDD y traversing the DAG eginning at the root node. At eah node X, if the variale orresponding to the index of X is true, we selet the T-Child. Otherwise we selet with the E-Child. Eventually we will reah either 0 or 1, indiating the value of the funtion on this input. We augment this with the Zero-Suppression Rule: we may eliminate nodes whose T Child is 0. With these standard rules, 0 represents the empty olletion of sets, while 1 represents the olletion onsisting of only the empty set. ZDDs interpreted this way have a standard set of operations ased on reursive definitions [13], inluding the union and intersetion of two olletions of sets, for example. 3 A Motivating Example a µ ßÞ Ð 1 µ ßÞ Ð 2 d eµ ßÞ Ð 3 ā µ ßÞ Ð 4 d eµ ßÞ Ð Steps Taken y Breadth-First Searh In general, BFS expands nodes in its queue until reahing a violated lause, at whih point the searh spae is pruned. As a result, the numer of nodes grows quikly. The initial pass of the BFS onsiders the first variale a. Both a 0 and a 1 are valid partial truth assignments sine neither violates any lauses. BFS ontinues y enqueueing oth of these partial truth assignments. In Figure 2, the ontents of the BFS Queue are listed as it-strings. At Step 2, the BFS onsiders oth possile values for. Beause of the lause a µ, BFS determines that a 0, 0 is not a valid partial truth assignment. The remaining three partial truth assignments are valid, and BFS enqueues them. At Step 3, eause of the lause ā µ we know that a,, and annot all e true. The searh spae is pruned further eause µ removes all ranhes involving 1 0. At Step 4, the state spae doules.

5 a Front: a Front: a Front: {{1}, {4}} {{2}, {}, {2, 4}} {{}} {1} {4} {2} {} {2, 4} {} 0 1 a Front: a {{}, {5}} {} {5} Step 1 Step 2 Step 2a Step 3 Step3a {} Front: {{}} Fig. 3. Steps Taken y Cassatt 3.2 Steps Taken y Cassatt The Cassatt algorithm performs a similar searh, however partial truth assignments are not stored expliitly. As a result, if two partial truth assignments orrespond to the same set of open lauses then only one of these will e stored. While the algorithm is desried in Setion 5, here we demonstrate possile redutions in the searh spae. We will all Cassatt s olletion of sets of open lauses the front. Variale a appears in two lauses: lause 1, a µ and lause 4, ā µ. If the variale a is assigned a 1, then lause 1 eomes satisfied while making lause 4 open. If a 0, then lause 1 eomes open. Beause we annot yet determine whih of these assignments (if any) will lead to to a satisfying truth assignment, it is neessary to store oth ranhes. The front shown in Figure 3, Step 1, is updated to ontain the two sets of open lauses whih orrespond to the two possile valid partial truth assignments. Variale appears in three lauses: lause 1, a µ, lause 2, µ, and lause 4, ā µ. These lauses are our new set of ativated lauses. For the first set, 1, we onsider oth possile truth assignments, 0 and 1. Assigning 0 satisfies lauses 2 and 4. However, it does not satisfy lause 1. We also see that orresponds to the end literal for lause 1. If we do not hoose the value of whih satisfies this lause, it will never e satisfied. As a result we annot inlude this ranh in the new front. Note that realizing this only depends on notiing that orresponds to the end literal for some open lause. For 1 we see that lause 1 is satisfied, ut lauses 2 and 4 are not yet satisfied. So, 2 4 should e in the front after this step is ompleted. A similar analysis an e performed for the remaining set in the front, 4. In Cassatt it is not neessary to onsider these sets in suession; ZDD algorithms allow us to perform these operations on multiple sets simultaneously. These sets of lauses, listed in the front shown in Step 2, are all the state information Cassatt maintains. It is here that this algorithm egins to differ sustantially from a naive BFS. One of the sets in the front is the empty set. This means that there is some assignment of variales whih ould satisfy every lause in whih any of these variales appear. As a result, there is no need to examine any other truth assignment; the partial truth assignment whih leaves no open lauses is learly and provaly the est hoie. As a result of storing sets of open lauses, Cassatt prunes the searh spae and only onsiders the single node.

6 Whereas the traditional BFS needs to onsider three alternatives at this point, Cassatt, y storing open lauses, dedues immediately that one of these is provaly superior. As a result, only a single node is expanded. As seen in Step 2a, this effetively restrutures the searh spae so that several of the possile truth assignments are susumed into a single ranh. As Cassatt proesses the third variale,, it needs to only onsider its effets on the single, empty suset of open lauses. Sine eah new variale ativates more lauses, this empty suset one again, an potentially grow. Again, one possile variale hoie satisfies all ativated lauses. This is shown in Step 3. As a result, Cassatt an one again susume multiple truth assignments into a single ranh as indiated in Step 3a. 4 Advantages of Storing Open Clauses If two partial truth assignments reate the exat same set of open lauses, then learly spae an e saved y only storing one of these susets. This also uts the searh spae. Moreover, if a stored set of lauses A is a proper suset of another set of lauses B, then the partial truth assignment orresponding to B is su-optimal and an e disarded. This is so eause every lause whih is affeted y any partial truth assignment thus far (every ativated lause) has een onsidered when produing A and B. As a result, there are no onsequenes to hoosing the truth assignment whih produes A over the truth assignment whih produes B. Every satisfying truth assignment ased on the partial truth assignment giving B orresponds to a satisfying truth assignment ased on A. These susumption rules are naturally addressed y the ZDD data struture, whih appears oth appropriate and effetive as a ompat data struture. We use three ZDD operations originally introdued in [4]. The susumed differene X A X B is defined as the olletion of all sets in X A whih are not susumed y some set in X B. Based on the susumed differene operator, it is possile to define an operator NoSu Xµ as the olletion of all sets in X whih are not susumed y some other set in X. Finally, the susumption-free union X A ËS X B is defined as the olletion of sets in X A Ë XB from whih all susumed sets have een removed. For the sake of revity, we will not repeat the full, reursive definitions of these operators here, ut only point out that the operators introdued in [4] were shown for slightly different ZDD semantis to enode olletions of lauses. Here we use the original ZDD semantis proposed y Minato [13] to enode olletions of susets. However, the definitions of operators themselves are unaffeted. These operators provide a mehanism for maintaining a susumption-free olletion of sets. ZDDs are known to often ahieve good ompression when storing sparse olletions. In general a olletion of sets may or may not ompress well when represented y ZDDs. However in Cassatt our representation an e pruned y removing sets whih are susumed y another set. Intuitively, this leads to a sparser representation for two reasons: sets with small numers of elements have a greater hane of susuming larger sets in this olletion, and eah susumption redues the numer of sets whih need to e stored. By using operators whih eliminate susumed sets, then, we hope to improve the hane that our given olletion will ompress well. In Cassatt this orresponds to improved memory utilization and runtime.

7 A natural question to ask is whether the numer of susets to e stored (the front) has an upper ound. An easy ound ased on the utwidth is otained y noting that for lauses, the maximum numer of inomparale susets is exatly the size of the maximal anti-hain of the partially ordered set 2. Theorem 4.1 [10, p. 444] The size of the maximal anti-hain of 2 is given y 2. This simple upper ound does not take into aount the details of Cassatt. Doing this may yield a sustantially tighter ound. It should e lear from the example that simply y storing sets of open lauses, Cassatt potentially searhes a muh smaller spae than a traditional BFS. In general, it annot fully avoid the explosion in spae. Although there are potentially many more ominations of lauses than there are variales, Cassatt annot examine more nodes than a straightforward BFS. At eah stage, Cassatt has the potential to redue the numer of nodes searhed. Even if no redution is ever possile, Cassatt will only examine as many nodes as a traditional BFS. 5 Compressed Breadth-First Searh The Cassatt algorithm implements a readth-first searh ased on keeping trak of sets of open lauses from the given CNF formula. The olletion of these sets represents the state of the algorithm. We refer to this olletion of sets as the front. The algorithm advanes this front just as a normal readth-first algorithm searhes suessively deeper in the searh spae. The ritial issue is how the front an e advaned in a readth-first manner and determine satisfiaility of a CNF formula. To explain this, we will first e preise aout what should happen ased on omining a single set of open lauses with a truth assignment to some variale v. We will then show how this an e done to the olletion of sets y using standard operations on ZDDs. Given a set of open lauses orresponding to a valid partial truth assignment t, and some assignment to a single variale v r, there are two main steps whih must e explained. First, Cassatt determines if the omination of t with v r produes a new valid partial truth assignment. Then, Cassatt must produe the new set of open lauses orresponding to the omination of t with v r. Sine the front orresponds to the olletion of valid partial truth assignments, then if this front is ever empty (equal to 0), then the formula is unsatisfiale. If Cassatt proesses all variales without the front eoming empty then no lauses will e ativated, ut unsatisfied. Thus if the formula is satisfiale, the front will ontain only the empty set (equal to 1) at the end of the searh. 5.1 Deteting Violated Clauses With respet to the variale ordering used, literals within a lause are divided into three ategories: eginning, middle, and end. The eginning literal is proessed first in the variale ordering while the end literal is proessed last. The remaining literals are alled middle literals. A nontrivial lause may have any (nonnegative) numer of middle literals. However, it is useful to guarantee the existene of a eginning and an end literal.

8 If a lause does not have at least 2 literals, then it fores the truth assignment of a variale, and the formula an e simplified. This is done as a preproessing step in Cassatt. Cassatt knows in advane whih literal is the end literal for a lause, and an use this information to detet violated lauses. Reall that a violated lause ours when all the variales whih appear within a lause have een assigned a truth value, yet the lause remains unsatisfied. Thus only the end literal has the potential to reate a violated lause. The value whih does not satisfy this lause produes a onflit or violated lause. This is the ase sine the sets stored in the front ontain only unsatisfied lauses. After proessing the end variale, no assignment to the remaining variales an affet satisfiaility. Determining if the omination of t and v r is a valid partial truth assignment is thus equivalent to onsidering if v orresponds to an end literal for some lause in the set orresponding to t. If the assignment v r auses this literal to e f alse then all literals in are false, and omining t with v r is not valid. 5.2 Determining the New Set of Open Clauses Given a set of lauses S orresponding to some partial truth assignment and an assignment to a single variale v, all lauses in S whih do not ontain any literal orresponding to v s should, y default, propagate to the new set S ¼. This is true, sine any assignment to v does not affet these lauses in any way. Consider a lause not in S ontaining v as its eginning variale. If the truth assignment given to v auses this lause to e satisfied, then it should not e added to S ¼. Only if the truth assignment auses the lause to remain unsatisfied does it eome an open lause and should e added to S ¼. Finally if a lause in S ontains v as one of its middle variales then the literal orresponding to this variale must again e examined. If the truth assignment given to v auses this lause to eome satisfied, then it should not e propagated to S ¼. Otherwise, it should propagate from S to S ¼. Tale 1. New-Suset Rules Beginning Middle End None Satisfied Impossile No Ation No Ation No Ation Open Impossile if(t lµ 0) No Ation S ¼ S Ë ¼ C S ¼ S Ë ¼ C Unassigned if(t lµ 0) Impossile Impossile No Ation S ¼ S Ë ¼ C These rules are summarized in tale 1. For a given lause and an assignment to some variale, the ation to e taken an e determined y where the literal appears in the

9 lause (olumn) and the urrent status of this lause (row). Here, No Ation means that the lause should not e added to S ¼. It should e noted that when an end variale appears in an open lause, no ation is neessary only with regard to S ¼ (it does not propagate into S ¼ ). If the assignment does not satisfy this lause, then the partial truth assignment is invalid. However, given that a partial truth assignment is valid then these lauses must eome satisfied y that assignment. 5.3 Performing Multiple Operations via ZDDs Cassatt s ehavior for a single set was desried in Tale 1. However, the operations desried here an e reformulated as operations on the entire olletion of sets. By simply iterating over eah set of open lauses and performing the operations in Tale 1, one ould ahieve a orret, if ineffiient implementation. However, it is possile to represent the omined effet on all sets in terms of ZDD operations. In order to reate an effiient implementation, we perform these ZDD operations on the entire front rather than iterating over eah set. To illustrate this, onsider how some of the lauses in eah suset an e violated. Reall that if a lause is violated, then the set ontaining annot lead to satisfiaility and must e removed from the front. The same violated lauses will appear in many sets, and eah set in the front ontaining any violated lause must e removed. Instead of iterating over eah set, we interset the olletion of all possile sets without any violated lause with the front; all sets ontaining any violated lause will e removed. Suh a olletion an e formulated using ZDDs. More ompletely, a truth assignment t to a single variale v has the following effets: It violates some lauses. Let U v t e the set of all lauses suh that the end literal of orresponds to v and is false under t. Then U v t is the set of lauses whih are violated y this variale assignment. It satisfies some lauses. Let S v t e the set of all lauses suh that the literal of orresponding to v is true under t. If these lauses were not yet satisfied, then they eome satisfied y this assignment. It opens some lauses. Let A v t e the set of all lauses suh that the eginning literal of orresponds to v and is false under t. Then A v t is the set of lauses whih have just een ativated, ut remain unsatisfied. Note that eah of these sets depends only on the partiular truth assignment to v. With eah of these sets of lauses, an appropriate ation an e taken on the entire front. 5.4 Updating Based on Newly Violated Clauses Given a set of lauses U v t whih have just een violated y a truth assignment, all susets whih have one or more of these lauses must e removed from the front.

10 Eah suset ontaining any of these lauses annot yield satisfiaility. We an uild the olletion of all sets whih do not ontain any of these lauses. This new olletion of sets will have a very ompat representation when using ZDDs. Let Uv t ¼ denote the olletion of all possile sets of lauses whih do not ontain any elements from U v t. That is, S ¾ U ¼ v t S Ì U v t /0. The ZDD orresponding to U ¼ v t is shown in Figure 4. In Figure 4, the dashed arrow represents a node s E- Child while a solid arrow represents a node s T-Child. Beause of the Zero-Suppression rule, any node whih does not appear in the ZDD is understood not to appear in any set in the olletion. As a result, to reate a ZDD without lauses in U v t, no element in U v t should appear as a node in the ZDD for Uv t ¼. Beause we look at a finite set of lauses C, we wish to represent the remaining lauses C ÒU v t as don t are nodes in the ZDD. To form the don t are nodes, we simply reate a ZDD node with oth its T- hild and its E-hild pointing to the same suessor. The resulting ZDD has the form shown in Figure 4a. When we u 1 u 2 u 3 u n 1 a 1 a 2 a 3 a n 0 1 Fig. 4: (a) The U ¼ v t ZDD. () The A ¼ v t ZDD take front front Ì Uv t ¼ we will get all susets whih do not ontain any lauses in U v t. As a result, the front will e pruned of all partial truth whih eome invalid as a result of this assignment. Instead of inluding all lauses in C ÒU v t, we an further redue this operation y inluding only lauses within the ut. Inluding nodes to preserve lauses outside the ut is superfluous. The ZDD representing Uv t ¼ an thus e reated ontaining O utwidthµ nodes in the worst ase. 5.5 Updating Based On Newly Satisfied Clauses Consider a lause whih has just een satisfied and appears somewhere in the front. Sine the front onsists of sets of ativated, unsatisfied lause, then it must have een open (unsatisfied) under some partial truth assignment. However, this ourrene has just een satisfied. As a result, every ourrene of a satisfied lause an simply e removed entirely from the front. for eah ¾ S v t do Temp 0 Suset0(Front, ) Temp 1 Suset1(Front, ) Front Temp 0 Ë Temp1 Fig. 5: Pseudoode for astration Among other operations, the standard operations on ZDDs inlude ofatoring, or Suset0 and Suset1. Suset0(Z, a) reates the olletion of all sets in the ZDD Z without a given element a, while Suset1(Z, a) reates the olletion of all sets in a ZDD Z whih did ontain a given element a. However, the element a is not present in any of the ZDDs returned y either Suset0 or Suset1. As a result, these operations an e used to eliminate a given element from the front. This idea is illustrated in the pseu-

11 doode in Figure 5. The omined effet of these operations on a ZDD is alled existential astration [14] and an e implemented with a single operation on the ZDD [15]. 5.6 Updating Newly Opened Clauses Consider a set of newly opened lauses A v t. Eah set in the front should now also inlude every element in A v t. We use the ZDD produt to form the new front y letting A ¼ v t A v t and finding front A ¼ v t. When we uild the ZDD A ¼ v t, only those lauses in A v t need to appear as nodes in A ¼ v t. This is eause of the Zero-Suppression rule: nodes not appearing in the ZDD A ¼ v t must e 0. The ZDD for A ¼ v t is simple as it only ontains a single set. The form of all suh ZDDs is shown in Figure 4. When the ZDD produt operation is used, front A ¼ v t will form the new front. In many ases, a shortut an e taken. By properly numering lauses with respet to a given variale ordering, at eah step we an ensure that eah lause index is lower than the minimum index of the front. In this ase, all lauses should e plaed aove the urrent front. The front will essentially remain unhanged, however new nodes in the form of Figure 4 will e added parenting the root node of the front. In this ase, the overhead of this operation will e linear in the numer of opened lauses. 5.7 Compressed BFS Pseudoode CASSATT(Vars, Clauses) Front 1 for eah v ¾ Vars do Front Front Form sets U v 0 U v 1 S v 0 S v 1 A v 0 A v 1 Build the ZDDs U ¼ v 0 U ¼ v 1 Front Front Ì U ¼ v 1 Front Front Ì U ¼ v 0 Front Astrat(Front, S v 1 ) Front Astrat(Front, S v 0 ) Build the ZDDs A ¼ v 0 A¼ v 1 Front Front A ¼ v 1 Front Front A ¼ v 0 Front Front Ë S Front if Front == 0 then return Unsatisfiale if Front == 1 then return Satisfiale Fig. 6: Pseudoode for the Cassatt Algorithm Now that we have shown how the front an e modified to inlude the effets of a single variale assignment, the operation of the omplete algorithm should e fairly lear. For eah variale v, we will opy the front at a given step, then modify one opy to reflet assigning v 1. We will modify the other opy to reflet assigning v 0. The new front will e the union with susumption of these two. ZDD nodes are in reality managed via referene ounts. When a ZDD (or some of its nodes) are no longer needed, we derement this referene ount. When we opy a ZDD, we simply inrement the referene ount of the root node. Thus, opying the front in Cassatt takes only onstant overhead. The pseudoode of the Cassatt algorithm is shown in Figure 6.

12 Here, we order the steps of the algorithm aording to the MINCE ordering, applied as a separate step efore Cassatt egins. We initially set the front to the olletion ontaining the empty set. This is onsistent, eause trivially there are no open lauses yet. Reall that after all variales are proessed, either the front will ontain only the empty set (e 1), or will e the empty olletion of sets (e 0). 6 Empirial Validation We implemented Cassatt in C++ using the CUDD pakage [18]. We also used an existential astration routine from the Extra lirary [15]. For these results, we disaled reordering and garage olletion. These tests were performed on an AMD Athlon 1.2GHz mahine, with 1024MB of 200MHz DDR RAM running Deian Linux. We also inlude performane of Chaff [16] and GRASP [17], two leading DLL-ased SAT solvers. We also inlude results of ZRes, a solver whih performs the DP proedure [7] while ompressing lauses with a ZDD. All solvers were used with their default onfigurations, exept ZRes whih requires a speial swith when solving hole-n instanes. All runs were set to time out after 500s. We ran two sets of enhmarks whih are onsidered to e diffiult for traditional SAT solvers [3]. The hole-n family of enhmarks are ome from the pigeonhole priniple. Given n 1 pigeons and n holes, the pigeonhole priniple implies that least 2 pigeons must live in the same hole. The hole-n enhmarks are a CNF enoding of the negation of this priniple, and are unsatisfiale as they plae at most 1 pigeon in eah hole. The numer of lauses in this family of enhmarks grows as Θ n 3 µ while the numer of variales grows as Θ n 2 µ. For hole-50, there are 2550 variales and lauses. The Urquhart enhmarks are a randomly generated family relating to a lass of prolems ased on expander graphs [20]. Both families have een used to prove lower ounds for runtime of lassial DP and DLL algorithms and ontain only unsatisfiale instanes. [3] shows that any DLL or resolution proedure requires Ω 2 n 20 µ time for hole-n instanes. Figure 7 empirially demonstrates that Cassatt requires Θ n 4 µ time for these instanes. This does not inlude time used to generate the MINCE variale ordering or I/O time. The runtimes of the four solvers tested on instanes from hole-n are graphed in Figure 7a. The smallest instane we onsider is hole-5; Cassatt effiiently proves unsatisfiaility of hole-50 in under 14 seonds. Figure 7 plots the runtimes of Cassatt on hole-n vs. n for n 50 on a log-log sale. We also inlude a plot of n 4 with a onstant set to learly show the relationship with the empirial data (on the log-log sale, this effetively translates the plot vertially). The Urquhart-n instanes have some degree of randomness. To help ompensate for this, we tested 10 different randomly generated instanes for eah n, and took the average runtime of those instanes whih ompleted. Memory appears to e the limiting fator for Cassatt, rather than runtime; it annot solve one Urq-8 instane, while quikly solving others. With respet to the DIMACS enhmark suite, Cassatt effiiently solves many families. Most aim enhmarks are solved, as well as pret, and duois instanes. However, DLL solvers outperform Cassatt on many enhmarks in this suite.

13 seonds holes Cassatt Chaff ZRes GRASP seonds holes Cassatt n**4/ Fig. 7. (a) Runtime of four SAT solvers on hole-n. () Cassatt s runtime on holen. Cassatt ahieves asymptoti speedup over three other solvers Tale 2. Benhmark times and ompletion ratios for Urquhart-n. Timeouts were set to 500s Urq-n Average Cassatt Chaff [16] ZRes [5] Grasp [17] # Vars # Clauses % Solved Avg % Solved Avg % Solved Avg % Solved Avg %.07s 50% 139s 100%.20s 0% %.39s 0% - 100%.57s 0% % 1.32s 0% - 100% 1.35s 0% % 4.33s 0% - 100% 2.83s 0% % 43.27s 0% - 100% 5.39s 0% % 43.40s 0% - 100% 9.20s 0% - 7 Conlusions In this paper, we have introdued the Cassatt algorithm, a ompressed BFS whih solves the SAT deision prolem. It leverages the ompression power of ZDDs to ompat state representation. Also, the novel method for enoding the state uses susuming semantis of ZDDs to redue redundanies in the searh. In addition, the runtime of this algorithm is dependent on the size of the ompressed, impliit representation rather than on the size of an expliit representation. This is the first work to attempt using a ompressed BFS as a method for solving the SAT deision prolem. Our empirial results show that Cassatt is ale to outperform DP and DLL ased solvers in some ases. We show this y examining runtimes for Cassatt and omparing this to proven lower ounds for DP and DLL ased proedures. Our ongoing work aims to add Boolean Constraint Propagation to the Cassatt algorithm. With Constraint Propagation, any unsatisfied lause whih has only one remaining unassigned variale fores the assignment of that variale. It is hoped that taking this into aount will give a redution in memory requirements and runtime. Another diretion for future researh is studying the effets of SAT variale and BDD orderings on the performane of the proposed algorithm.

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