An Exponential Separation between Deterministic CDCL and DPLL Solvers

An Exp onential Sepa ration b et w een Deterministic CDCL and DPLL Solvers Sahil Samar # Sc ho ol of Computer Science, Georgia Institute of T ec hnology , USA Marc Vin yals # Sc ho ol of Computer Science, Univ ersit y of A uc kland, New Zealand Vija y Ganesh # Sc ho ol of Computer Science, Georgia Institute of T ec hnology , USA Abstract W e prov e that there exists a deterministic configuration of Conflict Driven Clause Learning (CDCL) SA T solvers using a v ariant of the VSIDS branc hing heuristic that solves instances of the Ordering Principle (OP) CNF form ulas in time p olynomial in n , where n is the n umber of v ariables in such form ulas. Since tree-lik e resolution is kno wn to hav e an exp onential lo wer bound for pro of size for OP formulas, it follows that CDCL under this configuration has an exponential separation with an y solver that is polynomially equiv alent to tree-lik e resolution and therefore an y configuration of DPLL SA T solvers. 2012 ACM Subject Classification Theory of computation → Logic Keyw o rds and phrases SA T Solvers, Proof Systems, VSIDS 2 An Exponential Sepa ration betw een Deterministic CDCL and DPLL Solvers 1 Intro duction Empirically sp eaking, it is w ell-known that practical SA T solv ers based on the CDCL algorithm are efficient for certain classes of problems [10]. It is also well-kno wn that SA T solvers based on CDCL are muc h more efficient than their DPLL coun terparts, and that the main reason for that is clause learning. This is supp orted b oth exp erimen tally , as shown in ablation studies [ 11 , 8 ], as well as theoretically , given how DPLL is constrained b y the very w eak tree-lik e resolution pro of system, while CDCL escap es that constraint. Not only CDCL is p oten tially stronger than tree-like resolution, there is an equiv alence b et w een algorithms and pro of systems. Assuming b oth algorithms employ optimal heuristics, DPLL b ecomes equiv alent to tree-like resolution, while CDCL b ecomes equiv alent to general resolution [ 15 , 2 ]. Since there exist form ulas that hav e p olynomially long resolution pro ofs but require exp onen tially long tree-lik e resolution pro ofs, as a result of the separation b etw een pro of systems we obtain an exp onential separation betw een algorithms. Ho w ever, there is a cav eat to the equiv alence and separation results, which is that they only apply to nondeterministic algorithms using optimal heuristics giv en by an unrealistic oracle. The fact that a short pro of exists do es not mean that a deterministic algorithm is necessarily able to find it. Whether that is p ossible is called the automatability problem [ 6 ], and it is an active area of study . In the case of resolution, a breakthrough result pro v ed that no deterministic algorithm can alwa ys find short resolution pro ofs unless P = NP [3]. T o b e a bit more precise, there is only one heuristic that is required to b e nondeterministic for CDCL to simulate resolution, and that is the branching heuristic, which determines the next v ariable to b e branched up on. A randomized heuristic is enough for CDCL to simulate b ounded-width resolution, and if the heuristic is fixed and deterministic then it is known unconditionally—indep enden tly of whether P = NP —that CDCL cannot alw ays find short resolution pro ofs when the branching heuristic is VSIDS-lik e [18] or ordered [14]. Therefore, we are still left with the question of explaining the p ow er of CDCL with practical branching heuristics suc h as VSIDS, and in particular whether there exists an exp onen tial separation b etw een DPLL and CDCL in a practical setting, and with practical heuristics. In other words , whether there are form ulas that cannot b e solved unless CDCL exploits the full p ow er of resolution. W e address this question here. T w o families of formulas that exp onen tially separate tree-like and general resolution, and therefore are candidates to answ er our question, are p ebbling formulas and the ordering principle. The first family of formulas has b een studied in a pro of system more closely resem bling CDCL [ 9 ], but still assuming an optimal decision heuristic. The second family is what concerns us. The complexity of resolution pro ofs of the ordering principle has been widely studied. The formulas w ere in tro duced as a tricky family that could b e solved using certain symmetry breaking prop erties but app eared hard for resolution [ 12 ]. They were later shown to hav e in fact short resolution pro ofs [ 17 ] but to require exp onentially long tree-like resolution pro ofs [ 5 ]. F urthermore, a mo dified version of these formulas w as used to exp onentially separate regular from general resolution [1]. What makes these formulas tricky is that any resolution pro ofs require learning clauses con taining man y v ariables, which is usually an indicator of hardness. In a well-defined sense, these formulas are just at the b oundary of formulas that are easy v ersus hard for resolution: a form ula o ver n v ariables that requires clauses con taining Ω( √ n ) v ariables to pro ve, suc h as the ordering principle, requires exp onentially long tree-lik e resolution pro ofs, and a formula whose S. Sama r, M. Viny als, and V.Ganesh 3 pro of requires clauses containing Ω( n 1 / 2+ ϵ ) v ariables requires exp onentially long resolution pro ofs. This mak es the ordering principle a very interesting case study for the capabilities of an y resolution-based algorithm. F urthermore, empirical work gives strong evidence that the Glucose SA T Solver with VSIDS deca y factor 0.6 can solv e OP instances in p olynomial time, but not with a larger deca y factor of 0.95 [8], further adding to the m ystery . In this pap er, we prov e that CDCL with either the VSIDS or VMTF heuristics solves the ordering principle family of form ulas in p olynomial time. T o the b est of our knowledge, this is the first result sho wing a p olynomial upp er bound for a completely deterministic configuration of CDCL solvers, simultaneously exp onentially separating it from all configurations of DPLL SA T solvers. T o b e more precise, we use VSIDS with decay factor at most 1 / 2 (which is equiv alen t to a v ariant of VMTF), fixed phase v alue selection, restarts after ev ery conflict, no clause deletion, and 1UIP clause learning. Our pro of hea vily uses the fact that the deca y factor is at most 1 / 2 , which is p erhaps not surprising in view of how an aggressive decay factor has also b een experimentally observ ed to b e required to solv e instances of the ordering principle. Most assumptions we mak e ab out the deterministic configuration of CDCL studied here are inspired by practice or fairly standard in theoretical pap ers, except p erhaps fixed phase v alue selection. F or example, frequent restarts and no deletions are required in the existing pro ofs that nondeterministic CDCL sim ulates resolution, while 1UIP is the de-facto standard clause learning heuristic in mo dern CDCL SA T solvers. An immediate corollary of our result, given that there is an exp onen tial lo wer b ound on the size of tree-like resolution pro ofs of the ordering principle, and that the DPLL algorithm is captured by tree-like resolution, is that there is an exp onential separation b et w een deterministic CDCL and nondeterministic DPLL. 2 Prelimina ries In this section, we briefly outline the definitions used in the rest of this pap er. 2.1 Ordering Principle A n = ^ 1 ≤ i,j,k ≤ n i  = j,k  = i,j  = k ( ¬ P i,j ∨ ¬ P j,k ∨ P i,k ) , ( transitivit y) B n = ^ 1 ≤ i,j ≤ n i  = j ( ¬ P i,j ∨ ¬ P j,i ) , (an tisymmetry) D n = ^ 1 ≤ j ≤ n _ 1 ≤ i ≤ n i  = j P i,j ( non-minimalit y ) A ( i, j, k ) = ¬ P i,j ∨ ¬ P j,k ∨ P i,k ( clause in A n ) B ( i, j ) = ¬ P i,j ∨ ¬ P j,i ( clause in B n ) D ( j ) = _ 1 ≤ i ≤ n i  = j P i,j ( clause in D n ) An Ordering Principle instance O P n is defined as A n ∧ B n ∧ D n . In particular, the v ariables in the formula are P i,j s.t. 1 ≤ i, j ≤ n and i  = j . F or a literal P i,j , we often will call i the ro w and j the column. An O P n instance can b e though t of as describing n elemen ts that 4 An Exp onential Separation b et w een Deterministic CDCL and DPLL Solvers are ordered in some wa y where P i,j enco des that element i is smaller than element j , and suc h that the elemen ts satisfy transitivity and antisymmetry conditions, but also with the prop ert y that no element is minimal (and thus, the formula is UNSA T). 2.2 VSIDS First in tro duced b y the authors of the Chaff solver [ 13 ], VSIDS (V ariable State Indep enden t Deca ying Sum) is a dynamic branching heuristic for SA T Solv ers that, up on learning a conflict clause, assigns a score to each v ariable that app ears in the conflict clause (or some v ariation of the implication graph), decays the scores of all v ariables at regular interv als, and then branches on the highest scoring v ariable not curren tly on the assignment trail. There ha v e b een many v ariants of the heuristic since its first in tro duction. The v ariant of VSIDS that we consider is identical to that used in the first version of MiniSA T [ 7 ], i.e., it scores variables not literals, bumps the v ariables in the final learned clause, rather than all or some subset of clauses in volv ed in conflict analysis, and decays all v ariable scores by a multiplicativ e factor after e ach c onflict . There is a single hyperparameter for the heuristic, whic h is the deca y factor δ . The scores are all initialized to 0 , and up dated after each conflict. Let the score of v ariable x after conflict t b e q ( x, t ) . Then, the up date formula is: q ( x, t ) = b ( x, t ) + δ q ( x, t − 1) where b ( x, t ) = 1 if v ariable x participated in conflict t and b ( x, t ) = 0 otherwise. F or our pro of, we define “participated in conflict” as b eing a part of the resulting learned clause of the conflict. It is also well known that VSIDS with decay factor at most 1 / 2 is equiv alen t to the V ariable Mov e to F ron t (VMTF) branc hing heuristic [ 16 , 4 ]. Th us, we can replace VSIDS with VMTF for this pro of, and the argument would work the same. 2.3 CDCL Configuration The SA T Solver configuration we analyze in this pap er is the follo wing: VSIDS Branc hing heuristic (as describ ed in the previous section) with decay factor δ ≤ 1 / 2 Fixed phase v alue selection: when making decisions, alwa ys assign v ariables to false. Restart after every conflict No clause deletion scheme 1UIP Clause Learning scheme W e denote the database of learned clauses by Γ . W e assume that at any p oint during the solv er run, Γ contains all of the clauses the solv er learned up to that p oint in the same order in which they were learned. The only detail left to sp ecify is how w e break ties when there is more than one literal with the highest VSIDS score. Since w e are sp ecifically analyzing OP instances, we define the tie-break rule as follows: Let the v ariables P i,j of the OP instance b e in column ma jor order, i.e. P 2 , 1 , ..., P n, 1 , P 1 , 2 , ..., P n, 2 , ..., P 1 ,n , ..., P n − 1 ,n When branc hing, if there is a tie for the highest VSIDS score, we break the tie based on this ordering (i.e. the v ariable chosen is the one with the highest score and whichev er comes first in the ab ov e ordering). S. Sama r, M. Viny als, and V.Ganesh 5 3 Main Result W e sho w that CDCL can determine the unsatisfiability of the ordering principle in polynomial time b y a direct analysis of the b eha vior of the algorithm. That is, for eac h clause that the algorithm learns we sho w whic h decisions and unit propagations the solv er do es until reac hing a conflict, building an implication graph along the w ay , and argue that our construction is complete. Then we show which clause the conflict analysis heuristic learns when applied to said implication graph. W e first prov e a useful inv ariant ab out the VSIDS branching heuristic as defined in the CDCL configuration describ ed ab ov e. ▶ Lemma 1 ( F o cus Lemma) . A ssume that the solver has le arne d at le ast one clause. Consider the set S of variables in the most r e c ent le arne d clause. During a run, when the solver has to br anch, it must br anch on an unassigne d variable in S and assign it to false, b efor e it c an br anch on any other variables. Pro of. First, observe that the v ariables in the most recen tly learned clause ha ve a VSIDS score of at least 1 . The reason is that their scores are increased by a v alue of 1 after the clause was learned by the definition of VSIDS. V ariables not in the most recent learned clause ha v e VSIDS score at most P ∞ i =1 δ i = δ 1 − δ . Observe that δ 1 − δ < 1 for δ < 1 2 . F urthermore, for any finite k ∈ N , P k i =1 1 2 i < 1 . So, the v ariables in the most recen t learned clause all ha v e higher VSIDS score than any v ariable not in the most recent learned clause for δ ≤ 1 / 2 . Therefore, up on branching, the solver branches on unassigned v ariables in the new est learned clause first, b efore branching on an y other v ariables. Finally , b ecause of fixed phase v alue selection, the solver assigns the (unassigned) v ariables in the most recent learned clause to false. ◀ Informally , what the F ocus Lemma says is that whenever the solv er has to make a decision during a run, the solver cannot branch on a v ariable whic h do es not appear in the newest learned clause if ther e exists at le ast one unassigne d variable in the newest le arne d clause . Note that it could b e the case that a v ariable in the most recent learned clause is already propagated to true b efore a conflict is deriv ed, and the solv er thus didn’t assign false to it. The F o cus Lemma says nothing ab out this. A dditionally , observe that the F ocus Lemma can be extended b ey ond just the newest learned clause. Under this configuration, we can view branc hing as ranking each v ariable based solely on the most recent learned clause it app ears in (where higher rank means the v ariable app eared in a newer learned clause), and then picking the v ariable with highest rank (and using the tie-breaking rule if necessary). F or the following, consider any OP instance O P n for n ≥ 6 1 , and CDCL configured as describ ed in the previous section. ▶ Definition 2 ( Ordered Literal Sequence) . F or any c olumn j , let L j b e the or der e d se quenc e of variables in that c olumn: L j = ⟨ P 1 ,j , P 2 ,j , . . . , P j − 1 ,j , P j +1 ,j , . . . , P n − 1 ,j , P n,j ⟩ Note that for j = 1 , v ariable P 1 , 1 do es not exist, so the sequence starts at P 2 , 1 . ▶ Definition 3 ( Prefix Clause) . L et C ( j, k ) b e the disjunction of the first k variables in L j . 1 W e need this condition because the T ail do es not exist for n < 6 6 An Exp onential Separation b et w een Deterministic CDCL and DPLL Solvers ▶ Theo rem 4. The solver always le arns exactly the fol lowing clauses in this exact or der on O P n instanc es: He ad C (1 , n − 2) C (1 , n − 3) Desc ending Casc ade F or e ach j desc ending fr om n − 2 to 2 : C ( j, n − 2) C ( j, n − 3) . . . C ( j, j − 1)          A triangular blo ck of depth n − j . Starts at length n − 2 , ends at length j − 1 . T ail C (1 , n − 4) . . . C (1 , 2)      R esumes c olumn 1. Ends with the p air P 2 , 1 ∨ P 3 , 1 . T o illustrate the pattern, for OP 6 (left) and O P 7 (righ t) the learned clauses are: P 2 , 1 P 3 , 1 P 4 , 1 P 5 , 1 P 2 , 1 P 3 , 1 P 4 , 1 P 1 , 4 P 2 , 4 P 3 , 4 P 5 , 4 P 1 , 4 P 2 , 4 P 3 , 4 P 1 , 3 P 2 , 3 P 4 , 3 P 5 , 3 P 1 , 3 P 2 , 3 P 4 , 3 P 1 , 3 P 2 , 3 P 1 , 2 P 3 , 2 P 4 , 2 P 5 , 2 P 1 , 2 P 3 , 2 P 4 , 2 P 1 , 2 P 3 , 2 P 1 , 2 P 2 , 1 P 3 , 1 P 2 , 1 P 3 , 1 P 4 , 1 P 5 , 1 P 6 , 1 P 2 , 1 P 3 , 1 P 4 , 1 P 5 , 1 P 1 , 5 P 2 , 5 P 3 , 5 P 4 , 5 P 6 , 5 P 1 , 5 P 2 , 5 P 3 , 5 P 4 , 5 P 1 , 4 P 2 , 4 P 3 , 4 P 5 , 4 P 6 , 4 P 1 , 4 P 2 , 4 P 3 , 4 P 5 , 4 P 1 , 4 P 2 , 4 P 3 , 4 P 1 , 3 P 2 , 3 P 4 , 3 P 5 , 3 P 6 , 3 P 1 , 3 P 2 , 3 P 4 , 3 P 5 , 3 P 1 , 3 P 2 , 3 P 4 , 3 P 1 , 3 P 2 , 3 P 1 , 2 P 3 , 2 P 4 , 2 P 5 , 2 P 6 , 2 P 1 , 2 P 3 , 2 P 4 , 2 P 5 , 2 P 1 , 2 P 3 , 2 P 4 , 2 P 1 , 2 P 3 , 2 P 1 , 2 P 2 , 1 P 3 , 1 P 4 , 1 P 2 , 1 P 3 , 1 Pro of. The structure of our pro of is to sho w that the solv er first learns the Head , then transitions to and learns the Descending Cascade , and finally transitions to and learns the T ail and deriv es UNSA T. W e inductiv ely show that, given the solv er has learned some prefix of the learned clauses in the theorem statement, it correctly learns the next clause. S. Sama r, M. Viny als, and V.Ganesh 7 W e show the corresp onding implication graphs to learned clauses, and for eac h graph w e sho w the 1UIP no de and cut. First, we state a k ey inv ariant: ▶ Lemma 5 ( Equal Score In v ariant) . F or the given CDCL solver c onfigur ation, when solving O P instanc es, the variables in a le arne d clause C al l have the same VSIDS sc or e right after the clause C has b e en le arne d and the sc or es have b e en bump e d. Note that, for some clause, the Equal Score In v ariant holds in t wo obvious cases: (1) if that clause consists entirely of v ariables which don’t app ear in any previous learned clauses, and (2) if the v ariables in the learned clause form a subset of the set of v ariables in a prior learned clause, those v ariables didn’t app ear in any other learned clauses in b etw een, and the Equal Score Inv ariant held for that prior clause. W e will show that, for eac h learned clause, it falls into one of these tw o cases. A consequence of the Equal Score Inv arian t is that we branc h on literals in the most recent learned clauses (by the F o cus Lemma) in column-ma jor order (due to the tie-breaking rule). 1 : A (2 , n, 1) 2 : D (1) 3 : D (1) 4 : A ( n − 1 , n, 1) 5 : A (2 , n, 1) 6 : A ( n − 1 , n, 1) 7 : B (1 , n ) 8 : D ( n ) 9 : D ( n ) 10 : D ( n ) ¬ P 2 , 1 @1 ¬ P n − 1 , 1 @ n − 2 P n, 1 @ n − 2 ¬ P 2 ,n @ n − 2 ¬ P n − 1 ,n @ n − 2 ¬ P 1 ,n @ n − 2 CONFLICT . . . . . . 1 2 3 4 5 6 7 8 9 10 Figure 1 First Head Clause Implication Graph. 1UIP no de and cut in red. A t first, since | Γ | = 0 and all VSIDS scores are 0 , the solver branches on v ariables according to the column ma jor enco ding of OP; that is, the solv er branc hes on P 2 , 1 , P 3 , 1 , ..., P n − 1 , 1 (and assigns them to false). After doing so, w e get a conflict, and 1UIP deriv es the learned clause C (1 , n − 2) (Figure 1). Since the scores were all 0 b efore, clearly the Equal Score Inv arian t holds by Case 1. Then, by the F o cus Lemma, the solver branches on P 2 , 1 , ..., P n − 2 , 1 b efore 8 An Exp onential Separation b et w een Deterministic CDCL and DPLL Solvers an y other v ariables. This then yields a conflict and 1UIP derives the learned clause C (1 , n − 3) (Figure 2). Because each literal in this learned clause was also in the previous learned clause, the Equal Score In v ariant is main tained. So, w e see that the learned clause database alw a ys starts off with Head , and the Equal Score In v ariant is main tained for the Head b y Case 2. 1 : A (2 , n − 1 , 1) 2 : C (1 , n − 2) 3 : C (1 , n − 2) 4 : A ( n − 2 , n − 1 , 1) 5 : B (1 , n − 1) 6 : A (2 , n − 1 , 1) 7 : A ( n − 2 , n − 1 , 1) 8 : A (1 , n, n − 1) 9 : D ( n − 1) 10 : A (2 , n, n − 1) 11 : D ( n − 1) 12 : D ( n − 1) 13 : A ( n − 2 , n, n − 1) 14 : A (1 , n, n − 1) 15 : A (2 , n, n − 1) 16 : A ( n − 2 , n, n − 1) 17 : B ( n − 1 , n ) 18 : D ( n ) 19 : D ( n ) 20 : D ( n ) 21 : D ( n ) ¬ P 2 , 1 @1 . . . ¬ P n − 2 , 1 @ n − 3 P n − 1 , 1 @ n − 3 ¬ P 2 ,n − 1 @ n − 3 . . . ¬ P n − 2 ,n − 1 @ n − 3 ¬ P 1 ,n − 1 @ n − 3 P n,n − 1 @ n − 3 ¬ P 1 ,n @ n − 3 ¬ P 2 ,n @ n − 3 . . . ¬ P n − 2 ,n @ n − 3 ¬ P n − 1 ,n @ n − 3 CONFLICT 2 1 3 4 6 7 5 11 10 12 13 9 8 14 15 16 17 18 19 20 21 Figure 2 Second Head Clause Implication Graph F or the Descending Cascade , there are three facts that we show. The first is that the Descending Cascade actually b e gins , i.e., we learn C ( n − 2 , n − 2) right after the head. The second is that we c asc ade within a single column, i.e., given that the solver has learned C ( j, n − 2) , ..., C ( j, k ) for k > j − 1 , the solver learns C ( j, k − 1) next (Lemma 6). The third is that after learning C ( j, j − 1) for j > 2 , we desc end to the next column and learn S. Sama r, M. Viny als, and V.Ganesh 9 C ( j − 1 , n − 2) (Lemma 7). After learning the Head, by the F o cus Lemma, we see that the solver next branc hes on P 2 , 1 , ..., P n − 3 , 1 , P n − 1 , 1 . After doing so, 1UIP derives the learned clause C ( n − 2 , n − 2) (Figure 3). None of the literals in the learned clause were inv olved in an y of the previous learned clauses, so the Equal Score In v ariant holds b y case 1. 1 : A (2 , n − 2 , 1) 2 : C (1 , n − 3) 3 : C (1 , n − 3) 4 : A ( n − 3 , n − 2 , 1) 5 : A ( n − 1 , n − 2 , 1) 6 : B (1 , n − 2) 7 : A (2 , n − 2 , 1) 8 : A ( n − 3 , n − 2 , 1) 9 : A ( n − 1 , n − 2 , 1) 10 : A (1 , n, n − 2) 11 : D ( n − 2) 12 : A (2 , n, n − 2) 13 : D ( n − 2) 14 : D ( n − 2) 15 : A ( n − 3 , n, n − 2) 16 : D ( n − 2) 17 : A ( n − 1 , n, n − 2) 18 : A (1 , n, n − 2) 19 : A (2 , n, n − 2) 20 : A ( n − 3 , n, n − 2) 21 : A ( n − 1 , n, n − 2) 22 : B ( n − 2 , n ) 23 : D ( n ) 24 : D ( n ) 25 : D ( n ) 26 : D ( n ) 27 : D ( n ) ¬ P 2 , 1 @1 ¬ P n − 3 , 1 @ n − 4 ¬ P n − 1 , 1 @ n − 3 . . . P n − 2 , 1 @ n − 4 ¬ P 1 ,n − 2 @ n − 4 ¬ P 2 ,n − 2 @ n − 4 ¬ P n − 3 ,n − 2 @ n − 4 ¬ P n − 1 ,n − 2 @ n − 3 . . . P n,n − 2 @ n − 3 ¬ P 1 ,n @ n − 3 ¬ P 2 ,n @ n − 3 ¬ P n − 3 ,n @ n − 3 ¬ P n − 1 ,n @ n − 3 . . . ¬ P n − 2 ,n @ n − 3 CONFLICT 1 2 3 4 5 6 7 8 9 11 10 14 15 16 17 12 13 18 19 20 21 22 23 24 25 26 27 Figure 3 First Descending Cascade Clause Implication Graph 10 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers ▶ Lemma 6 ( Cascade Lemma) . A ssume that Γ c ontains the He ad clauses fol lowe d by C ( j, n − 2) , ..., C ( j, j − 1) for j = n − 2 , ..., l + 1 , and then C ( l, n − 2) , ..., C ( l, k ) for k > l − 1 . A dditional ly, assume that the Equal Sc or e Invariant holds for al l of those clauses. The next le arne d clause is C ( l , k − 1) , and the Equal Sc or e Invariant stil l holds. There are 4 cases for the Cascade Lemma, shown b elow. 1 : A (1 , k + 1 , l ) 2 : C ( l , k ) 3 : A ( l − 1 , k + 1 , l ) 4 : C ( l , k ) 5 : C ( l , k ) 6 : A ( l + 1 , k + 1 , l ) 7 : C ( l , k ) 8 : A ( k , k + 1 , l ) 9 : A (1 , k + 1 , l ) 10 : A ( l − 1 , k + 1 , l ) 11 : A ( l + 1 , k + 1 , l ) 12 : A ( k , k + 1 , l ) 13 : B ( l, k + 1) 14 : A (1 , k + 2 , k + 1) 15 : C ( k + 1 , k ) 16 : A ( l − 1 , k + 2 , k + 1) 17 : C ( k + 1 , k ) 18 : C ( k + 1 , k ) 19 : A ( l + 1 , k + 2 , k + 1) 20 : C ( k + 1 , k ) 21 : A ( k , k + 2 , k + 1) 22 : C ( k + 1 , k ) 23 : A ( l, k + 2 , k + 1) 24 : A (1 , k + 2 , k + 1) 25 : A ( l − 1 , k + 2 , k + 1) 26 : A ( l, k + 2 , k + 1) 27 : A ( l + 1 , k + 2 , k + 1) 28 : A ( k , k + 2 , k + 1) 29 : B ( k + 1 , k + 2) 30 : C ( k + 2 , k + 1) 31 : C ( k + 2 , k + 1) 32 : C ( k + 2 , k + 1) 33 : C ( k + 2 , k + 1) 34 : C ( k + 2 , k + 1) 35 : C ( k + 2 , k + 1) ¬ P 1 ,l @1 . . . ¬ P l − 1 ,l @ l − 1 ¬ P l +1 ,l @ l . . . ¬ P k,l @ k − 1 P k +1 ,l @ k − 1 ¬ P 1 ,k +1 @ k − 1 . . . ¬ P l − 1 ,k +1 @ k − 1 ¬ P l +1 ,k +1 @ k − 1 . . . ¬ P k,k +1 @ k − 1 ¬ P l,k +1 @ k − 1 P k +2 ,k +1 @ k − 1 ¬ P 1 ,k +2 @ k − 1 . . . ¬ P l − 1 ,k +2 @ k − 1 ¬ P l,k +2 @ k − 1 ¬ P l +1 ,k +2 @ k − 1 . . . ¬ P k,k +2 @ k − 1 ¬ P k +1 ,k +2 @ k − 1 CONFLICT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Figure 4 Cascade Lemma ( k  = l, k = n − 2 ) Implication Graph S. Sama r, M. Viny als, and V.Ganesh 11 1 : A (1 , k + 1 , l ) 2 : C ( l , k ) 3 : C ( l , k ) 4 : A ( l − 1 , k + 1 , l ) 5 : A (1 , k + 1 , l ) 6 : A ( l − 1 , k + 1 , l ) 7 : B ( l, k + 1) 8 : C ( k + 1 , k ) 9 : C ( k + 1 , k ) 10 : C ( k + 1 , k ) ¬ P 1 ,l @1 . . . ¬ P k − 1 ,l @ k − 1 P k +1 ,l @ k − 1 ¬ P 1 ,k +1 @ k − 1 . . . ¬ P l − 1 ,k +1 @ k − 1 ¬ P l,k +1 @ k − 1 CONFLICT 1 2 3 4 5 6 7 8 9 10 Figure 5 Cascade Lemma ( k = l, k  = n − 2 ) Implication Graph 1 : A (1 , k + 1 , l ) 2 : C ( l , k ) 3 : A ( l − 1 , k + 1 , l ) 4 : A ( k − 1 , k + 1 , l ) 5 : C ( l , k ) 6 : A ( l + 1 , k + 1 , l ) 7 : C ( l , k ) 8 : A ( k , k + 1 , l ) 9 : A (1 , k + 1 , l ) 10 : A ( l − 1 , k + 1 , l ) 11 : A ( l + 1 , k + 1 , l ) 12 : A ( k , k + 1 , l ) 13 : B ( l, k + 1) 14 : C ( k + 1 , k ) 15 : C ( k + 1 , k ) 16 : C ( k + 1 , k ) 17 : C ( k + 1 , k ) 18 : C ( k + 1 , k ) ¬ P 1 ,l @1 . . . ¬ P l − 1 ,l @ l − 1 ¬ P l +1 ,l @ l . . . ¬ P k,l @ k − 1 P k +1 ,l @ k − 1 ¬ P 1 ,k +1 @ k − 1 . . . ¬ P l − 1 ,k +1 @ k − 1 ¬ P l +1 ,k +1 @ k − 1 . . . ¬ P k,k +1 @ k − 1 ¬ P l,k +1 @ k − 1 CONFLICT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Figure 6 Cascade Lemma ( l < k < n − 2) Implication Graph 12 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers 1 : A (1 , k + 1 , l ) 2 : C ( l , k ) 3 : C ( l , k ) 4 : A ( k − 1 , k + 1 , l ) 5 : A (1 , k + 1 , l ) 6 : A ( k − 1 , k + 1 , l ) 7 : B ( k , k + 1) 8 : A (1 , k + 2 , k + 1) 9 : C ( k + 1 , k + 1) 10 : C ( k + 1 , k + 1) 11 : A ( k − 1 , k + 2 , k + 1) 12 : C ( k + 1 , k + 1) 13 : A ( k , k + 2 , k + 1) 14 : A (1 , k + 2 , k + 1) 15 : A ( k − 1 , k + 2 , k + 1) 16 : A ( k , k + 2 , k + 1) 17 : B ( k + 1 , k + 2) 18 : C ( k + 2 , k + 1) 19 : C ( k + 2 , k + 1) 20 : C ( k + 2 , k + 1) 21 : C ( k + 2 , k + 1) ¬ P 1 ,l @1 . . . ¬ P k − 1 ,l @ k − 1 P k +1 ,l @ k − 1 ¬ P 1 ,k +1 @ k − 1 . . . ¬ P k − 1 ,k +1 @ k − 1 ¬ P k,k +1 @ k − 1 P k +2 ,k +1 @ k − 1 ¬ P 1 ,k +2 @ k − 1 . . . ¬ P k − 1 ,k +2 @ k − 1 ¬ P k,k +2 @ k − 1 ¬ P k +1 ,k +2 @ k − 1 CONFLICT 1 2 3 4 5 6 7 8 9 10 12 11 13 14 17 15 16 18 21 19 20 Figure 7 Cascade Lemma ( k = l = n − 2 ) Implication Graph The Equal Score Inv ariant holds b y Case 2 for the Cascade Lemma. ▶ Lemma 7 ( Descending Lemma) . A ssume that Γ c ontains the He ad clauses fol lowe d by C ( j, n − 2) , ..., C ( j, j − 1) for j = n − 2 , ..., l , for l > 2 . A dditional ly, assume that the Equal Sc or e Invariant holds for al l of those clauses. Then, the next le arne d clause is C ( l − 1 , n − 2) , and the Equal Sc or e Invariant stil l holds. Due to the size of the implication graph, it is left to the App endix; see Figure 10. The Equal Score Inv ariant holds by Case 1 for the Descending Lemma. W e no w mov e on to the T ail . W e first notice that the last learned clause in the Descending Cascade is alwa ys the unit clause C (2 , 1) = P 1 , 2 . There are tw o things we need to sho w. The first is that the T ail b egins, i.e., we learn C (1 , n − 4) after learning P 1 , 2 (Lemma 8). The second is that after learning C (1 , n − 4) , ..., C (1 , k ) for k > 2 , the next learned clause is C (1 , k − 1) (Lemma 9). ▶ Lemma 8 ( Piv ot Lemma) . A ssume that Γ c ontains the He ad clauses fol lowe d by the Desc ending Casc ade clauses, the last of which is the unit clause P 1 , 2 . A dditional ly, assume that the Equal Sc or e Invariant holds for al l of those clauses. The next le arne d clause is C (1 , n − 4) , and the Equal Sc or e Invariant stil l holds. S. Sama r, M. Viny als, and V.Ganesh 13 1 : P 1 , 2 2 : A (3 , 1 , 2) 3 : A ( n − 3 , 1 , 2) 4 : A (3 , 1 , 2) 5 : A ( n − 3 , 1 , 2) 6 : A (2 , n − 2 , 1) 7 : C (1 , n − 3) 8 : A (3 , n − 2 , 1) 9 : C (1 , n − 3) 10 : C (1 , n − 3) 11 : A ( n − 3 , n − 2 , 1) 12 : A (2 , n − 2 , 1) 13 : A (3 , n − 2 , 1) 14 : A ( n − 3 , n − 2 , 1) 15 : B (1 , n − 2) 16 : A ( l, n, l − 1) 17 : A ( l + 1 , l − 1 , l ) 18 : A ( n − 1 , l − 1 , l ) 19 : C ( n − 2 , n − 3) P 1 , 2 @0 ¬ P 3 , 2 @1 . . . ¬ P n − 3 , 2 @ n − 5 ¬ P 2 , 1 @0 ¬ P 3 , 1 @1 . . . ¬ P n − 3 , 1 @ n − 5 P n − 2 , 1 @ n − 5 ¬ P 2 ,n − 2 @ n − 5 ¬ P 3 ,n − 2 @ n − 5 . . . ¬ P n − 3 ,n − 2 @ n − 5 ¬ P 1 ,n − 2 @ n − 5 CONFLICT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 8 Pivot Lemma Implication Graph The Equal Score Inv ariant holds b y Case 2 for the Pivot Lemma. ▶ Lemma 9 ( T ail Lemma) . A ssume that Γ c ontains the He ad clauses fol lowe d by the Desc ending Casc ade clauses fol lowe d by the clauses C (1 , n − 4) , ..., C (1 , k ) for k > 2 . A dditional ly, assume that the Equal Sc or e Invariant holds for al l of those clauses. The next le arne d clause is C (1 , k − 1) , and the Equal Sc or e Invariant stil l holds. 14 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers 1 : A ( k + 1 , 1 , 2) 2 : P 1 , 2 3 : A (2 , 1 , k + 1) 4 : C (1 , k ) 5 : A (3 , k + 1 , 1) 6 : C (1 , k ) 7 : C (1 , k ) 8 : A ( k , k + 1 , 1) 9 : A ( k + 1 , 1 , 2) 10 : A (2 , 1 , k + 1) 11 : A (3 , k + 1 , 1) 12 : A ( k , k + 1 , 1) 13 : B (1 , k + 1) 14 : C ( k , k + 1) 15 : C ( k , k + 1) 16 : C ( k , k + 1) 17 : C ( k , k + 1) P 1 , 2 @0 ¬ P 2 , 1 @0 ¬ P 3 , 1 @1 . . . ¬ P k, 1 @ k − 2 P k +1 , 1 @ k − 2 ¬ P 2 ,k +1 @ k − 2 ¬ P 3 ,k +1 @ k − 2 . . . ¬ P k,k +1 @ k − 2 ¬ P 1 ,k +1 @ k − 2 P k +1 , 2 @ k − 2 CONFLICT 1 2 6 5 7 8 9 4 3 10 11 12 13 14 15 16 17 Figure 9 T ail Lemma Implication Graph The Equal Score In v ariant holds by Case 2 for the T ail Lemma. The lemmas are enough to sho w that we learn the Head , Descending Cascade , and T ail clauses. T o complete the pro of of the theorem, we show that the CDCL solv er derives UNSA T up on learning these clauses. As part of the Descending Cascade , the unit clause P 1 , 2 is learnt. Propagating it, we derive ¬ P 2 , 1 . Then, the clause P 2 , 1 ∨ P 3 , 1 , which is in the T ail , b ecomes the unit clause P 3 , 1 . Then the transitivity clause A (2 , 3 , 1) b ecomes the unit clause ¬ P 2 , 3 . And, the an tisymmetry clause B (1 , 3) b ecomes the unit clause ¬ P 1 , 3 . W e then see that the clause P 1 , 3 ∨ P 2 , 3 , which is part of the Descending Cascade , is completely falsified. How ever, w e did not hav e to mak e any decisions to arrive at this conflict, and therefore we deriv e UNSA T. ◀ A natural question is whether we crucially need tie-breaking throughout the pro of. T o answ er this question we migh t try to generalize the pro of in a w ay so that the order in whic h we branch do es not matter, as long as our branching is column-fo cused (i.e. alw ays branc h in the same column as the previous decision if p ossible). This w ould result in p erhaps pro cessing columns in a different order but the pro of wou ld, plausibly , b e equiv alent up to symmetries. More precisely , we can assume that the first learned clause is C (1 , n − 2) , i.e. that we guarantee that w e start off column-fo cused; then, we ask if the F o cus Lemma and the rest of the configuration would result in the solver learning the essentially the same pro of without further in voking the tie-breaking rule. The answer is negative due to the Descending Lemma, the mechanism by which we transition columns in the Descending Cascade, whic h requires that w e initially propagate P l − 1 ,l from the first l − 2 decisions if w e wan t to learn a clause in column l − 1 . So, the previous learned clauses cannot consist of arbitrary subsets of literals from column l if w e wan t to descend to column l − 1 . This suggests that tie-breaking do es play an important role in this particular pro of, and that further generalizations that remo v ed tie-breaking would b e more complex if not longer. ▶ Theo rem 10. The total numb er of c onflicts for this deterministic c onfigur ation of CDCL SA T Solvers to derive UNSA T for an O P n instanc e is exactly  n 2  − 3 . S. Sama r, M. Viny als, and V.Ganesh 15 Pro of. The Head has 2 clauses, the Descending Cascade has P n − 2 j =2 n − j = n ( n − 3) / 2 clauses, and the T ail has n − 5 clauses. In total, then, there are ( n − 3)( n +2) 2 =  n 2  − 3 clauses. ◀ ▶ Co rollary 11. Ther e exists a deterministic CDCL solver that has an exp onential sep ar ation with any solver that is p olynomial ly e quivalent to tr e e-like r esolution (e.g. DPLL with nondeterministic br anching). Pro of. It is known that tree-like resolution has an exponential low er bound for pro of size for Ordering Principle instances [ 5 ]. W e hav e shown that CDCL under the analyzed configuration requires exactly  n 2  − 3 conflicts to derive UNSA T. It is w ell known that a p olynomial num b er of conflicts implies polynomial runtime for CDCL SA T Solv ers. It then follo ws that this CDCL solv er has an exp onential separation with an y solv er that is p olynomially equiv alent to tree-like resolution. ◀ 4 Conclusions In this pap er, we prov e that there exists an exp onential separation b etw een a deterministic configuration of CDCL SA T solv ers (that uses a v ariant of the VSIDS branching heuristic) and DPLL SA T solvers. More precisely , we sho w that the prop osed CDCL configuration can solv e the OP class of form ulas in p olynomial time, while it is known that tree-like resolution has exp onential low er b ounds for them. In the pro cess of proving our result, we identified some inv ariants of the VSIDS branc hing heuristics that may b e of broader interest. 16 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers References 1 Mic hael Alekhno vic h, Jan Johannsen, T oniann Pitassi, and Alasdair Urquhart. An exp onential separation b et w een regular and general resolution. The ory of Computing , 3(1):81–102, 2007. 2 Alb ert A tserias, Johannes Klaus Fich te, and Marc Th urley . Clause-learning algorithms with man y restarts and b ounded-width resolution. J. A rtif. Intel l. R es. , 40:353–373, 2011. URL: https://doi.org/10.1613/jair.3152 , doi:10.1613/JAIR.3152 . 3 Alb ert A tserias and Moritz Müller. Automating resolution is np-hard. J. A CM , 67(5):31:1– 31:17, 2020. doi:10.1145/3409472 . 4 Armin Biere and Andreas F röhlich. Ev aluating CDCL v ariable scoring schemes. In Marijn Heule and Sean A. 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In The Thirty- F ourth AAAI Confer enc e on A rtificial Intel ligenc e, AAAI 2020, The Thirty-Se c ond Innovative Applic ations of A rtificial Intel ligence Confer enc e, IAAI 2020, The T enth AAAI Symp osium on Educ ational A dvanc es in A rtificial Intel ligenc e, EAAI 2020, New Y ork, NY, USA, F ebruary 7-12, 2020 , pages 1652–1659. AAAI Press, 2020. URL: https://doi.org/10.1609/aaai. v34i02.5527 , doi:10.1609/AAAI.V34I02.5527 . 18 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers 5 App endix T o complete the pro of of Theorem 4, w e show that eac h of the implication graphs corres- p onding to the lemmas is c omplete . ▶ Definition 12 ( Completeness) . W e say that an implic ation gr aph is c omplete if it is imp ossible for a differ ent c onflict to b e le arne d under the same de cisions. W e use the following conv entions. A literal is derive d if it is either de cide d (the solver branc hes on the v ariable and assigns it a v alue) or if it is implie d by other literals. A literal l can only be implie d when, for some clause C s.t. l ∈ C , all literals except l ha v e their negation deriv ed. In that case, all literals in C \ l together imply l , and w e sa y that the clause C b ecame unit and l is the unit liter al . When a literal x implies a literal y via clause C , w e sa y that x pr op agates y b ecause of C . When a literal x is pr op agate d to satur ation , it means that for eac h clause that con tains x and is unit, we derive the unit literal. The c onflict clause is the clause that b ecomes completely falsified in the implication graph, and the le arne d clause is the clause learned by the solv er following conflict analysis (in these pro ofs, 1UIP). W e use these properties of BCP: ▶ Observation 13. A liter al is pr op agate d to satur ation b efor e any of the liter als that it implies ar e pr op agate d. ▶ Observation 14. Ther e ar e no further pr op agations onc e a c onflict has b e en identifie d. W e first define some general cases where clauses can b ecome unit. ▶ Observation 15 ( T ransitivit y Conditions) . R e c al l that tr ansitivity clauses ar e of the form A ( i, j, k ) = ¬ P i,j ∨ ¬ P j,k ∨ P i,k . Ther e ar e 3 c ases wher e the clause c ould b e c ome unit: 1. if you derive P i,j and P j,k 2. if you derive P j,k and ¬ P i,k 3. if you derive P i,j and ¬ P i,k A dditional ly, two liter als c an only (to gether) c ause a single T r ansitivity clause to b e c ome unit (fol lows simply fr om the fact that T r ansitivity clauses only have 3 liter als, and no two T r ansitivity clauses shar e mor e than 1 liter al). ▶ Observation 16 ( An tisymmetry Conditions) . R e c al l that antisymmetry clauses ar e of the form B ( i, j ) = ¬ P i,j ∨ ¬ P j,i . The only way this clause b e c omes unit is by deriving a liter al with p ositive p olarity. A liter al c an only c ause a single A ntisymmetry clause to b e c ome unit (fol lows simply fr om the fact that A ntisymmetry clauses only have 2 liter als, and no two A ntisymmetry clauses shar e mor e than 1 liter al). ▶ Observation 17 ( Non-minimalit y Conditions) . This only b e c omes unit when ther e ar e n − 2 liter als al l sharing the same c olumn and derive d with ne gative p olarity. ▶ Observation 18 ( Head Conditions) . The He ad clauses c an only b e c ome unit when ther e ar e n − 3 or n − 4 liter als al l with c olumn 1 and derive d with ne gative p olarity. ▶ Observation 19 ( Descending Cascade Conditions) . Desc ending Casc ade clauses c an only b e c ome unit when liter als within the same c olumn (notably not c olumn 1) ar e de cide d with ne gative p olarity. ▶ Observation 20 ( T ail Conditions) . T ail clauses c an only b e c ome unit when liter als with c olumn 1 ar e de cide d with ne gative p olarity. S. Sama r, M. Viny als, and V.Ganesh 19 ▶ Observation 21 ( Saturation Condition) . When a liter al l is pr op agate d to satur ation and the implie d liter als c ause ther e to b e a c onflict, none of the liter als that l implie d ar e pr op agate d. This is a dir e ct c onse quenc e of the BCP observations. 5.1 First Head Clause Completeness See Figure 1 for the implication graph. T ransitivit y: Eac h decision literal causes a T ransitivity clause to b ecome unit when P n, 1 is propagated to saturation. By the T ransitivit y Conditions, since tw o literals together can only make a single T ransitivit y clause unit, it follo ws that literals implied by P n, 1 m ust b e propagated in order for an y other T ransitivity clauses to b ecome unit. How ever, since propagating P n, 1 to saturation results in conflict, it follows by the Saturation Condition that no literal implied b y P n, 1 is propagated. Thus, no other T ransitivit y clause can b ecome unit. An tisymmetry: The only literal with p ositiv e polarity is P n, 1 , and the corresp onding an tisymmetry clause that becomes unit is shown. By the An tisymmetry Conditions, it follo ws that no other Antisymmetry clause could b ecome unit. Non-minimalit y: The o ccurrences of n − 2 literals sharing the same column with negative p olarit y are sho wn, and by the Non-minimalit y Conditions it follows that no other Non-minimalit y clause could b ecome unit. 5.2 Second Head Clause Completeness See Figure 2 for the implication graph. T ransitivit y: By Case 1 of the T ransitivit y Conditions, P n − 1 , 1 and P n,n − 1 together imply P n, 1 ; this is omitted from the implication graph b ecause P n, 1 cannot b e propagated by the Saturation Condition and it do es not app ear in the conflict clause. By the T ransitivity Conditions and the Saturation Condition, it follo ws that no other T ransitivity clause could b ecome unit. An tisymmetry: The only literals with p ositiv e p olarit y ha ve their corresp onding an tisym- metry clauses that b ecome unit sho wn. By the Antisymmetry Conditions, it follows that no other Antisymmetry clause could b ecome unit. Non-minimalit y: The o ccurrences of n − 2 literals sharing the same column with negative p olarit y are sho wn, and by the Non-minimalit y Conditions it follows that no other Non-minimalit y clause could b ecome unit. First Head Clause: This do es b ecome unit and is shown. 5.3 First Descending Cascade Clause Completeness See Figure 3 for the implication graph. By Case 1 of the T ransitivity Conditions, P n − 2 , 1 and P n,n − 2 together imply P n, 1 ; this is omitted from the implication graph because P n, 1 cannot b e propagated by the Saturation Condition and it do es not appear in the conflict clause. By the T ransitivit y Conditions and the Saturation Condition, it follo ws that no other T ransitivit y clause could b ecome unit. An tisymmetry: The only literals with p ositiv e p olarit y ha ve their corresp onding an tisym- metry clauses that b ecome unit sho wn. By the Antisymmetry Conditions, it follows that no other Antisymmetry clause could b ecome unit. 20 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers Non-minimalit y: The o ccurrences of n − 2 literals sharing the same column with negative p olarit y are sho wn, and by the Non-minimalit y Conditions it follows that no other Non-minimalit y clause could b ecome unit. Head clauses: The second Head clause b ecomes unit and is sho wn. The first Head clause cannot b ecome unit if the second Head clause b ecomes unit. 5.4 Cascade Lemma Completeness See Figures 4, 5, 6, 7 for the implication graph. T ransitivit y: Case 1 ( k  = l, k = n − 2 ): Shared ro w/col with p ositive p olarity (Case 1 of the T ransitivit y Conditions) has one p ossible o ccurrence which would propagate the literal P n,l as a result of A ( k + 2 , k + 1 , l ) ; this is omitted from the implication graph due to the Saturation Condition. Case 2 of the T ransitivity Conditions has all of its o ccurrences sho wn. Case 3 of the T ransitivit y Conditions do es not occur. Case 2 ( k = l, k  = n − 2) : It can easily b e seen that there is no other p ossible transitivity clause that could b ecome unit. Case 3 ( l < k < n − 2 ): It is clear that this is analogous to Case 2. Case 4 ( k = l = n − 2 : It is clear that this is analogous to Case 1. An tisymmetry: The only literals with p ositiv e p olarit y ha ve their corresp onding an tisym- metry clauses that b ecome unit sho wn. By the Antisymmetry Conditions, it follows that no other Antisymmetry clause could b ecome unit. Non-minimalit y: In all cases this cannot o ccur by the Non-minimality Conditions. Head clauses: W e never branch or derive any literals with column 1 so this cannot o ccur b y the Head Conditions. Descending Cascade Clauses: Case 1: W e derive literals with negative p olarity in column l , k + 1 , and k + 2 . All of the columns hav e their corresp onding Descending Cascade clauses shown to be unit. By the Descending Cascade Conditions, no other Descending Cascade clauses can b ecome unit. Case 2: W e derive literals with negative p olarit y in columns l and k + 1 , and the implication graph shows b oth of the corresp onding Descending Cascade clauses b ecome unit. By the Descending Cascade Conditions, no other Descending Cascade clauses can b ecome unit. Case 3: Analogous to Case 2. Case 4: Analogous to Case 1. S. Sama r, M. Viny als, and V.Ganesh 21 5.5 Descending Lemma Completeness ¬ P 1 ,l @1 . . . ¬ P l − 2 ,l @ l − 2 P l − 1 ,l @ l − 2 ¬ P 1 ,l − 1 @ l − 2 . . . ¬ P l − 2 ,l − 1 @ l − 2 ¬ P l,l − 1 @ l − 2 ¬ P l +1 ,l @ l − 1 . . . ¬ P n − 1 ,l @ n − 3 ¬ P l +1 ,l − 1 @ l − 1 . . . ¬ P n − 1 ,l − 1 @ n − 3 P n,l − 1 @ n − 3 ¬ P 1 ,n @ n − 3 . . . ¬ P l − 2 ,n @ n − 3 ¬ P l,n @ n − 3 ¬ P l +1 ,n @ n − 3 . . . ¬ P n − 1 ,n @ n − 3 ¬ P l − 1 ,n @ n − 3 P n,l @ n − 3 CONFLICT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Figure 10 Descending Lemma Implication Graph 22 An Exp onential Separation betw een Deterministic CDCL and DPLL Solvers 1 : A (1 , l − 1 , l ) 2 : C ( l , l − 1) 3 : A ( l − 2 , l − 1 , l ) 4 : C ( l , l − 1) 5 : A (1 , l − 1 , l ) 6 : A ( l − 2 , l − 1 , l ) 7 : A ( n, l − 1 , l ) 8 : B ( l − 1 , l ) 9 : A ( l + 1 , l − 1 , l ) 10 : A ( n − 1 , l − 1 , l ) 11 : A (1 , n, l − 1) 12 : D ( l − 1) 13 : A ( l − 2 , n, l − 1) 14 : D ( l − 1) 15 : D ( l − 1) 16 : A ( l, n, l − 1) 17 : A ( l + 1 , l − 1 , l ) 18 : A ( n − 1 , l − 1 , l ) 19 : D ( l − 1) 20 : A ( l + 1 , n, l − 1) 21 : D ( l − 1) 22 : A ( n − 1 , n, l − 1) 23 : A ( n, l − 1 , l ) 24 : A (1 , n, l − 1) 25 : A ( l − 2 , n, l − 1) 26 : A ( l, n, l − 1) 27 : A ( l + 1 , n, l − 1) 28 : A ( n − 1 , n, l − 1) 29 : B ( l − 1 , n ) 30 : D ( n ) 31 : D ( n ) 32 : D ( n ) 33 : D ( n ) 34 : D ( n ) 35 : D ( n ) Note that P n,l is not propagated due to the Saturation Condition. T ransitivit y: Shared ro w/col with p ositive p olarity (Case 1 of the T ransitivity Conditions) has one o ccurrence and is shown. Same column with opposite polarity (Case 2 of the T ransitivit y Conditions) also has all of its o ccurrences shown. Same ro w with opp osite p olarit y (Case 3 of the T ransitivit y Conditions) do es not occur. An tisymmetry: The only literals with p ositive p olarit y already hav e their corresp onding an tisymmetry clauses that b ecome unit shown. By the Antisymmetry Conditions, it follo ws that no other Antisymmetry clause could b ecome unit. Non-minimalit y: The only o ccurrences of this are shown, where D ( l − 1) b ecomes unit and D ( n ) is the conflict clause. By the Non-minimality Conditions, no other Non-minimalit y clause could b ecome unit. Head clauses: W e never branch or derive any literals with column 1 so this cannot o ccur b y the Head Conditions. Descending Cascade Clauses: The columns with literals branched on or propagated with negative polarity are l , l − 1 , and n . The Non-minimalit y clauses D ( l − 1) and D ( n ) become unit, so Descending Cascade clauses with literals in those columns cannot b ecome unit. The Descending Cascade clause C ( l, l − 1) is shown to b ecome unit. By the Descending Cascade Conditions, no other Descending Cascade clause can b ecome unit. 5.6 Pivot Lemma Completeness See Figure 8 for the implication graph. T ransitivit y: Case 1 of the T ransitivity Conditions o ccurs with P 1 , 2 and P n − 2 , 1 to imply P n − 2 , 2 , but is omitted from the implication graph since P n − 2 , 2 will not b e propagated due to the Saturation Condition and do es not app ear in the conflict clause. By the T ransitivit y Conditions and the Saturation Condition, no other T ransitivity clause can b ecome unit. An tisymmetry: The only literals with p ositive p olarit y already hav e their corresp onding an tisymmetry clauses that b ecome unit shown. By the Antisymmetry Conditions, it follo ws that no other Antisymmetry clause could b ecome unit. S. Sama r, M. Viny als, and V.Ganesh 23 Non-minimalit y: By the Non-minimality Conditions, no Non-minimalit y clauses can b ecome unit. Head clauses: The second Head clause b ecomes unit and is sho wn. The first Head clause cannot b ecome unit. Descending Cascade Clauses: The columns with literals branched on or propagated with negativ e polarity are 2 , 1 , and n − 2 . W e don’t consider 1 for the Descending Cascade; for column 2, since w e already learned the clause P 1 , 2 , all of the Descending Cascade Clauses with column 2 are already satisfied and cannot b ecome unit. W e falsify a Descending Cascade clause asso ciated with column n − 2 whic h is sho wn. By the Descending Cascade Conditions, no other Descending Cascade clauses can b ecome unit. 5.7 T ail Lemma Completeness See Figure 9 for the implication graph. T ransitivit y: Case 1 of the T ransitivity Conditions is only p ossible when we see literals suc h that one’s column is the same as the other’s row and they are derived with the same polarity , which happ ens with P 1 , 2 and P k +1 , 1 , from which we get P k +1 , 2 as sho wn. Case 2 of the T ransitivit y Conditions cannot o ccur apart from what is shown in the implication graph, b ecause w e never derive any other literals with the same column and opp osite p olarity . Finally , Case 3 of the T ransitivit y Conditions is only p ossible when we see literals such that they ha v e the same ro w and they are derived with opposite p olarity , whic h happ ens with P 1 , 2 and ¬ P 1 ,k +1 ; how ever, ¬ P 1 ,k +1 cannot b e propagated due to the Saturation Condition. An tisymmetry: The implication graph already sho ws all occurrences of this; P k +1 , 2 is not propagated b y the Saturation Condition. By the Antisymmetry Conditions, no other An tisymmetry clauses can b ecome unit. Non-minimalit y: No Non-minimalit y clauses can become unit b y the Non-minimalit y Conditions. Head: No Head clauses can b ecome unit by the Head Conditions. Descending Cascade: The only candidate is column k + 1 , whic h is the conflict clause in the implication graph. No other Descending Cascade clauses can b ecome unit by the Descending Cascade Conditions. T ail: The only tail clause whic h b ecomes unit is sho wn in the implication graph; no other tail clauses can b ecome unit b y the T ail Conditions.