Variable Independence and Resolution Paths for Quantified Boolean Formulas

Transcription

1 Variable Inepenence an Resolution Paths for Quantifie Boolean Formulas Allen Van Geler avg University of California, Santa Cruz Abstract. Variable inepenence in quantifie boolean formulas (QBFs) informally means that the quantifier structure of the formula can be rearrange so that two variables reverse their outer-inner relationship without changing the value of the QBF. Samer an Szeier introuce the stanar epenency scheme an the triangle epenency scheme to safely over-approximate the set of variable pairs for which an outer-inner reversal might be unsoun (JAR 2009). This paper introuces resolution paths an efines the resolution-path epenency relation. The resolution-path relation is shown to be the root (smallest) of a lattice of epenency relations that inclues quarangle epenencies, triangle epenencies, strict stanar epenencies, an stanar epenencies. Sounness is prove for resolution-path epenencies, thus proving sounness for all the escenants in the lattice. It is shown that the biconnecte components (BCCs) an block trees of a certain clause-literal graph provie the key to computing epenency pairs efficiently for quarangle epenencies. Preliminary empirical results on the 568 QBFEVAL-10 benchmarks show that in the outermost two quantifier blocks quarangle epenency relations are smaller than stanar epenency relations by wiely varying factors. 1 Introuction Variable inepenence in quantifie boolean formulas (QBFs) informally means that two variables that are ajacent in the quantifier structure can exchange places without changing the value of the QBF. The motivation for knowing such shifts are soun (i.e., cannot change the value of a close QBF, which is true or false) is that QBF solvers have more flexibility in their choice of which variable to select for a solving operation. They are normally constraine to obey the quantifier orer. Samer an Szeier introuce epenency schemes to recor epenency pairs (p,q) such that q is inner to p in the quantifier structure an any rearrangement that places q outer to p might be unsoun. The absence of (p,q) ensures that there is some soun rearrangement that places q outer to p [6]. The iea is that the pairs in a epenency scheme can be compute with reasonable effort, an are a safe over-approximation of the exact relation that enotes unsoun rearrangements of quantifier orer. A smaller epenency scheme allows

2 Resolution Path ¹ Quarangle ¹ Triangle ÈÈÈÈÈÈ Ê ÈÕ StrictStanar ¹ Stanar ¹ Trivial Fig. 1. Lattice of epenency relations. more pairs to be treate as inepenent. They propose two nontrivial schemes, the stanar epenency scheme, which is easiest to compute, but coarse, an the triangle epenency scheme, which is more refine. Lonsing an Biere have reporte favorable results on an implementation of the stanar epenency scheme [5]. We are not aware of any implementation of triangle epenencies. Lonsing an Biere provie aitional bibliography an iscussion of other approaches for increasing solver flexibility. This paper introuces resolution paths in Section 4 to efine a epenency relation that is smaller than those propose by Samer an Szeier. Resolution paths are certain paths in the resolution graph [7] associate with the quantifier-free part of the QBF. A hierarchy of new relations is introuce, calle resolution-path epenencies (smallest), quarangle epenencies, an strict stanar epenencies. Quarangle epenencies refine the triangle epenencies; strict stanar epenencies refine stanar epenencies. The resulting lattice is shown in Figure 1. Sounness is prove for resolution-path epenencies, thus proving sounness for all the escenants in the hierarchy. A slightly longer version of this paper contains some etails omitte here, ue to the page limit. 1 The main obstacle is computing the epenency relation for anything more refine than stanar epenencies or strict stanar epenencies. Samer an Szeier sketche a polynomial-time algorithm, which enable them to get interesting theoretical results involving triangle epenencies an back-oor sets. It appears to be too inefficient for practical use on large QBF benchmarks an, to the best of our knowlege, it has not been implemente. Samer an Szeier use a certain unirecte graph, similar to what is calle the clause-variable incience graph in the literature, for their algorithm. This clause-literal graph, as we shall call it, is normally alreay represente in the ata structures of a solver, as occurrence lists, an is practical to use for the stanar epenency relation [5]. It is easy to see stanar epenencies (an strict stanar epenencies) are base on the connecte components (CCs) of this graph. Strict stanar epenencies, introuce in Definition 5.2, are essentially a cost-free improvement on stanar epenencies, once this fact is recognize. This paper shows in Section 6 that the biconnecte components (BCCs) of the clause-literal graph provie the key to ientifying epenency pairs for 1 Please see avg/qbfeps/ for a more etaile version of this paper an a prototype program.

3 quarangle epenencies, introuce in Definition 5.2. Like CCs, BCCs can be compute in time linear in the graph size. Base on the BCC structure, the clause-literal graph can be abstracte into a block tree, so-calle in the literature. Quarangle epenencies can be etermine by paths in the block tree, which is normally much smaller than the clause-literal graph. Our algorithm coul be moifie to compute triangle epenencies, but this woul cost the same as quarangle epenencies, an prouce less inepenence, so this moification has not been implemente. We avoi calling the quarangle epenency relation a epenency scheme to avoi conflicting with the technical requirements state by Samer an Szeier [6] In a prototype C++ implementation that buils epenency relations, computing BCCs was foun to be as cheap as computing connecte components (neee for any epenency relation), on the 568 QBFEVAL-10 benchmarks. Preliminary empirical results are given in Section 7, mainly consisting of statistics about the BCC structure an size of quarangle epenency relations in these benchmarks. The primary goal of this work to provie methos by which practical QBF solvers can sounly carry out a broaer range of the operations they alreay perform. (Reaers shoul be familiar with QBF solver operations to follow these paragraphs, or come back after reaing Section 2.) The universal reuction operation is ubiquitous in QBF solvers. The stanar requirement is that all existential literals must be inepenent of the universal literal u to be elete in the trivial epenency relation. Theorem 4.9 shows that inepenence in the quarangle relation is sufficient. Search-base QBF solvers make variable assignments as assumptions (the wor ecision is often use). Normally, an existential variable can be selecte only if it is inepenent of all unassigne universal variables in the trivial epenency relation. Theorem 4.7 shows that inepenence in the quarangle relation is sufficient. 2 Preliminaries In general, quantifie boolean formulas (QBFs) generalize propositional formulas by aing universal an existential quantification of boolean variables. See [3] for a thorough introuction an a review of any unfamiliar terminology. A close QBF evaluates to either 0 (false) or 1 (true), as efine by inuction on its principal operator. 1. ( xφ(x)) = 1 iff (φ(0) = 1 or φ(1) = 1). 2. ( xφ(x)) = 0 iff (φ(0) = 0 or φ(1) = 0). 3. Other operators have the same semantics as in propositional logic. This efinition emphasizes the connection of QBF to two-person games, in which player E (Existential) tries to set existential variables to make the QBF evaluate to 1, an player A (Universal) tries to set universal variables to make the QBF evaluate to 0 (see [4] for more etails).

4 For this paper QBFs are in prenex conjunction normal form (PCNF), i.e., Ψ = Q. F consists of prenex Q an clause matrix F. Clauses may be written enclose in square brackets (e.g., [p,q, r ]). Literals are variables or negate variables, with overbar enoting negation. Usually, letters e an others near the beginning of the alphabet enote existential literals, while letters u an others near the en of the alphabet enote universal literals. Letters like p, q, r enote literals of unspecifie quantifier type. The variable unerlying a literal p is enote by p where necessary. The quantifier prefix is partitione into quantifier blocks of the same quantifier type. Each quantifier block has a unique qepth, with the outermost block having qepth = 1. The proof system known as Q-resolution consists of two operations, resolution an universal reuction. Q-resolution is of central importance for QBFs because it is a soun an complete proof system [2]. Resolution is efine as usual, except that the clashing literal is always existential; universal reuction is special to QBF. Let α, β, an γ be possibly empty sets of literals. res e (C 1,C 2 ) = α β where C 1 = [e,α], C 2 = [e, β] (1) unr u (C 3 ) = γ where C 3 = [C] 3 = [u,γ] (2) Resolvents must be non-tautologous for Q-resolution. unr u (C 3 ) is efine only if u is tailing for γ, which means that the quantifier epth (qepth) of u is greater than that of any existential literal in γ. A Q-erivation, often enote as π, is a irecte acyclic graph (DAG) in which each noe is either an input clause (a DAG leaf), or a proof operation (an internal noe) with a specifie clashing literal or reuction literal, an ege(s) to its operan(s). A Q-refutation is a Q-erivation of the empty clause. An assignment is a partial function from variables to truth values, an is usually represente as the set of literals that it maps to true. Assignments are enote by ρ, σ, τ, etc. Applications of an assignment σ to a logical expression are enote by q σ, C σ, F σ, etc. If σ assigns variables that are quantifie in Ψ, those quantifiers are elete in Ψ σ, an their variables receive the assignment specifie by σ. 3 Regular Q-Resolution In analogy with regular resolution in propositional calculus, we efine Q-resolution to be regular if no variable is resolve upon more than once on any path in the proof DAG. We nee the following property for analyzing resolution paths. Theorem 3.1 Regular Q-resolution an regular tree-like Q-resolution are complete for QBF. Proof: The proof for regular Q-resolution is the same as in the paper that showe Q-resolution is complete for QBF [2]. It is routine to transform a regular Q-resolution erivation into a regular tree-like Q-resolution erivation of the same clause, by splitting noes as neee, working from the leaves (original clauses) up.

5 4 Resolution Paths This section efines resolution paths an resolution-path epenencies, then states an proves the main results in Theorem 4.7 an subsequent theorems. Let a close PCNF Ψ = Q. G be given in which the quantifier block at qepth + 1 is existential. Consier the resolution graph G = (V,E) efine as follows [7]: Definition 4.1 The qepth-limite resolution graph G = (V, E) at qepth + 1 is the unirecte graph in which: 1. V, the vertex set, consists of clauses in G containing some existential literal of qepth at least + 1; 2. E, the unirecte ege set, consists of eges between clauses C i an C j in V, where there is a unique literal q such that q C i an q C j, so that C i an C j have a non-tautologous resolvent. Further, q is require to be existential an its qepth must be + 1 or greater. Each ege is annotate with the variable that qualifies it as an ege. A resolution path of epth + 1 is a path in G such that no two consecutive eges are annotate with the same variable. (Nonconsecutive eges with the same variable label are permitte an variable labels with qepths greater than + 1 are permitte.) Definition 4.2 We say that a literal p presses on an existential literal q in the graph G efine in Definition 4.1 if there is a resolution path of epth + 1 connecting a vertex that contains p with a vertex that contains q without using an ege annotate with q. Similarly, p presses on q if there is a resolution path of epth + 1 connecting a vertex that contains p with a vertex that contains q without using an ege annotate with q. One may think of presses on as a weak implication chain: if all the clauses involve are binary, it actually is an implication chain. An example is iscusse later in Example 5.4 an Figure 3 after some other graph structures have been introuce. The intuition is that if literal p presses on literal q, then making p true makes it more likely that q will nee to be true to make a satisfying assignment. Theorem 4.7 shows that transposing the variable orer in the quantifier prefix is soun, even though many combinations of pressing are present. Only certain combinations are angerous. We say that a sequence S is a subsequence of a sequence S if every element in S is also in S, in the same orer as S, but not necessarily contiguous in S. The next theorem shows that Q-resolution cannot bring together variables unless there is a presses on relationship in the original clauses. This suggests that resolution paths are the natural form of connection for variable epenencies. Theorem 4.3 Let Ψ = Q. G be a close PCNF. Let π be a regular tree-like Q-resolution erivation from Ψ. For all literals p an for all existential literals f, if there is a clause (input or erive) in π that contains both p an f, then the orer of sibling subtrees of π may be swappe if necessary so that a resolution

6 path from a clause with p to a clause with f appears as a subsequence of the leaves of π (not necessarily contiguous, but in orer). Proof: The proof is by inuction on the subtree structure of π. The base case is that p an f are together in a clause of G, say D 1, which is a leaf of π. Then D 1 constitutes a resolution path from p to f. For any non-leaf subtree, say π 1, assume the theorem hols for all proper subtrees of π 1. That is, assume for all literals q an for all existential literals e, if there is a clause in a proper subtree of π 1, say π 2, that contains both q an e, then the subtrees of π 2 may be swappe so that a resolution path from a clause with q to a clause with e appears as a subsequence of the leaves of π 2. Suppose that clause D 1, the root clause of π 1 contains both p an f. If p an f appear in a clause in a proper subtree of π 1, then the inuctive hypothesis states that the neee resolution path can be obtaine, so assume p an f o not appear together in any proper subtree of π 1. Arrange the two principal subtrees of π 1 so that p is in the root clause of the left subtree an f is in the root clause of the right subtree (p an/or f might be in both subtrees). Let the clashing literal be g at the root of π 1. That is, g appears in the left operan an g appears in the right operan of the resolution whose resolvent is D 1. By the inuctive hypothesis, the left subtree has a resolution path P L from a clause with p to a clause with g as a subsequence of its leaves. Also, the right subtree has a resolution path P R from a clause with g to a clause with f as a subsequence of its leaves. Concatenate P L an P R (with the ege being labele g ) to give a resolution path from a clause with p to a clause with f. Since g was a clashing literal at D 1, above the two subtrees, by regularity of the erivation, g cannot appear as an ege label in either P L or P R, so the concatenation cannot have consecutive eges labele with g. We now consier when transposing ajacent quantifie variables of ifferent quantifier types in the quantifier prefix oes not change the value of the QBF. Definition 4.4 Let a close PCNF Ψ = Q. G be given in which the universal literal u is at qepth an the existential literal e is at some qepth greater than. The pair (u, e) satisfies the resolution-path inepenence criterion if (at least) one of the following conitions hol in the epth-limite graph G efine in Definition 4.1: (A) u oes not press on e an u oes not press on e; or (B) u oes not press on e an u oes not press on e. If u an e are variables, the pair (u, e) satisfies the resolution-path inepenence criterion for variables if any of (u, e) or (u, e ) or (u, e) or (u, e ) satisfies the resolution-path inepenence criterion for literals. Definition 4.5 Let universal u an existential e be variables, as in Definition 4.4. We say the pair (u,e) is a resolution-path epenency tuple if an only if (at least) one of the following conitions hols in G: (C) u presses on e an u presses on e ; or

7 (D) u presses on e an u presses on e. Lemma 4.6 states that either this efinition or Definition 4.4, but not both, applies for pairs (u,e) of the correct types an qepths. Lemma 4.6 If u an e are universal an existential variables, respectively, then (u, e) satisfies the resolution-path inepenence criterion for variables if an only if e oes not have a resolution-path epenency upon u. Proof: Apply DeMorgan s laws an istributive laws to the efinitions. We are now reay to state the main theoretical results of the paper. We use transpose in its stanar sense to mean interchange of two ajacent elements in a sequence. Theorem 4.7 Let a close PCNF Ψ = Q. G be given in which the universal literal u is at qepth an is ajacent in the quantifier prefix to the existential literal e at qepth + 1. Let (u,e) satisfy the resolution-path inepenence criterion for literals (Definition 4.4). Then transposing u an e in the quantifier prefix oes not change the value of Ψ. Proof: It suffices to show that transposing u to a later position oes not cause Ψ to change in value from 1 to 0. We show this hols for all assignments σ to all variables outer to u in Ψ. That is, let Q rem be the suffix of Q beginning immeiately after u e, an efine Φ = u e Q rem. F, where F = G σ (3) Φ = e u Q rem. F. (4) Note that if the hypotheses (A) an (B) in Definition 4.4 hol for Ψ, then they also hol for Φ. Throughout this proof A an B refer to these conitions. Suppose Φ evaluates to 0. By Theorem 3.1 there is a regular tree-like Q-refutation π of Φ. Note that π has no reunant clauses; they all contribute to the refutation. Let us attempt to use π as a starter for π, which we want to be a Q-refutation of Φ. For notation, any prime symbol (such as D ) in Φ or π represents the corresponing unprime symbol (such as D) in Φ or π. What operation of π can be incorrect for π? The only possibilities are a universal reuction involving a clause containing literals on both u an e. In π, u is tailing w.r.t. e, whereas in π it is not. The key observation is that a regular tree-like Q-refutation erivation from Φ cannot prouce certain clauses containing literals on both u an e, ue to Theorem 4.3. Any resolution path in Φ from u or u to e or e that is implie by applying Theorem 4.3 to π cannot contain eges labele with e, by regularity. So such a path is also a resolution path after the transposition of u an e in the quantifier prefix. Such a resolution path in Φ or Φ is also a resolution path at the corresponing quantifier epth (i.e., + 1) in Ψ. The theorem hypothesis that Definition 4.4 hols, together with Lemma 4.6, prohibits certain resolution paths that woul imply that Definition 4.5 hols. As state, the only cases where the operation in π might not be imitate in π are where the operation is a universal reuction on u or u in a clause D. Let D in π correspon to D in π. Without loss of generality we assume that

8 all universals other than u or u have alreay been reuce out of D. There are several cases to examine, to show that the problematic operations in π can always be transforme into correct operations in π that achieve a Q-refutation of Φ. It will follow that transposing u an e oes not change the evaluation of Ψ. If D contains u, in π let the clause D 2 = unr u (D ). D 2 must contain e or e or the same reuction can apply to D. If D contains u an D 2 contains e, we cannot have case (B), so consier case (A). The reuce clause D 2 must resolve on e with some clause, say C, that contains e. But C cannot contain u. Let π resolve D with C, giving D 2. D 2 must be non-tautologous an now u can be reuce out, constructing a Q-refutation of Φ. If D contains u an D 2 contains e, neither case (A) nor case (B) is possible. If D contains u, in π let the clause D 3 = unr u (D ). D 3 must contain e or e or the same reuction can apply to D. If D contains u an D 3 contains e, D 3 must resolve with some clause, say C 3, that contains e. C 3 cannot contain u in either case (A) or (B). Let π resolve D with C 3, giving D 3. D 3 must be non-tautologous an now u can be reuce out, constructing a Q-refutation of Φ. If D contains u an D 3 contains e, we cannot have case (A), so consier case (B). The reuce clause D 3 must resolve with some clause, say C 4, that contains e. But C 4 cannot contain u. Let π resolve D with C 4, giving D 4. D 4 must be non-tautologous an now u can be reuce out, constructing a Q-refutation of Φ. Corollary 4.8 If e is an existential pure literal in the matrix of a close QBF Ψ, then e may be place outermost in the quantifier prefix without changing the value of Ψ. If u is a universal pure literal in a close QBF Ψ, then u may be place innermost in the quantifier prefix without changing the value of Ψ. Next we consier cases in which u an e are separate by more than one qepth. Although it might not be soun to revise the quantifier prefix, we still might be able to perform universal reuction an other operations sounly. Theorem 4.9 Let a close PCNF Ψ = Q. G be given in which the universal literal u is at qepth an the existential literals e 1,..., e k are at qepths greater than. Let C 0 = [α, u, e 1,...,e k ] be clause in G, where α (possibly empty) consists of existential literals with qepths less than an universal literals. For each i {1,...,k}, let ( u, e i ) satisfy the resolution-path inepenence criterion for variables (Definition 4.4). Then eleting u from C 0 oes not change the truth value of Ψ. That is, universal reuction on u in C 0 is soun. Proof: The proof iea is similar to Theorem 4.7, but is more involve because Theorem 4.3 nees to be invoke on multiple subtrees. It suffices to show that eletion of u from C 0 oes not cause Ψ to change from 1 to 0. We show this hols for all assignments σ to all variables outer to u in Ψ. That is, let Q rem be the suffix of Q beginning immeiately after u, an efine Φ = u Q rem. F, where F = G σ (5)

9 C2 ¼ g D ¼ 1 C1 ¼ u g C3 ¼ g 3 D ¼ 4 C4 ¼ g g 3 D ¼ 9 C8 ¼ e j Þ C ¼ Fig. 2. Refutation π exhibiting resolution path from u to e j for proof of Theorem 4.9. Circles contain clashing literals of resolutions that erive clauses immeiately above them. Φ = u Q rem. F, (6) where F is obtaine from F by replacing clause C = C 0 σ by C = C {u}. For notation, any prime symbol (such as D ) in Φ or π represents the corresponing unprime symbol (such as D) in Φ or π. Suppose Φ evaluates to 0. By Theorem 3.1 Φ has a regular tree-like Q- refutation, say π, which we use as a starter for π. The only operation in π that might be incorrect for π is a resolution involving a clause C 1 in π, where u C 1, u has been reuce out of C 1 in π, an the extra u causes the resolvent to be tautologous in π. Thus C 1 an C 1 contain at least one of the literals e 1,..., e k. Also C 1 an C 1 resolve with some clause D 1 = D 1 that contains u. We show this leas to a contraiction. Figure 2 shows the proof ieas. Let the resolvent of C 1 an D 1 in π be C 2 an let the clashing literal in D 1 be g. By Theorem 4.3 there is a resolution path from u to g using (some of) the leaves of the subtree roote at D 1. C provies a resolution-path from u to e i in Φ, for each i {1,...,k} so to establish the contraiction, it suffices to show that there is a resolution path from u to e j, for some j {1,...,k}. If g is equal to any of e 1,..., e k, we are one, so assume not. Swap the orer of sibling subtrees in π as necessary to place C on the rightmost branch, calle the right spline. Fin the lowest clause on this spline containing g. Call this clause C 3 an call its left chil D 4. D 4 contains g an the clashing literal use to erive C 3, say g 3. If D 4 contains e j for any j {1,...,k} rearrange its subtrees to exhibit a resolution path from g to e j an we are one. Otherwise, rearrange its subtrees to exhibit a resolution path from g to g 3, as suggeste in the figure. Appen this to the path from u to g (from the subtree eriving D 1), giving a resolution path from u to g 3. Continue extening the path in this manner own the right spline. That is, let C 5 be the lowest clause on this spline containing g 3 an let its left chil be D 6, etc. The figure oes not show these etails. Eventually, the left chil of a spline clause contains some e j, shown as C 8 in the figure. (This must occur at some point because the first resolution above C must use some e j as the

10 clashing literal.) When e j is reache, a resolution path from u to e j has been constructe, using the subtree that erives D 9 for the last segment. 5 Clause-Literal Graphs Let a close QBF Ψ be given in which the quantifier block at qepth + 1 is existential. We efine qepth-limite clause-literal graphs as follows: Definition 5.1 The qepth-limite clause-literal graph enote as G = ((V 0,V 1,V 2 ), E) at qepth +1 is the unirecte tripartite graph in which: The vertex set V 0 consists of clauses containing some existential literal of qepth at least + 1; The vertex set V 1 consists of existential positive literals of qepth at least + 1 that occur in some clause in V 0. The vertex set V 2 consists of existential negative literals of qepth at least + 1 that occur in some clause in V 0. The unirecte ege set E consists of (e i, e i ), where e i V 1, (e i,c j ), where e i V 1 an C j V 0 an e i C j, an (e i,c j ), where e i V 2 an C j V 0 an e i C j. See examples in Figure 3. Several epenency relations can be specifie in terms of paths in the epthlimite clause-literal graph G. Simple paths an simple cycles in G are efine as usual for unirecte graphs. Definition 5.2 Let u be a universal literal at qepth an let e be an existential literal at qepth + 1. A epenency pair ( u, e ) means e epens on u. 1. Stanar epenencies are base on connecte components. stdepa( u, e ) hols if any path in G connects a clause with universal literal u or u to a clause with existential literal e or e. 2. Strict stanar epenencies are base on connecte components of G. ssdepa( u, e ) hols if some path in G connects a clause with universal literal u to a clause with existential literal e or e, an some path in G connects a clause with u to a clause with e or e. 3. Quarangle epenencies are base on biconnecte components an articulation points of G, because they involve paths that avoi a certain literal. (Definitions are reviewe at the beginning of Section 6.) Articulation points are the only vertices that cannot be avoie. quadepa( u, e ) hols if; (A) Some path in G connects a clause with universal literal u to a clause with existential literal e an avois vertex e ; an (B) some path in G connects a clause with universal literal u to a clause with existential literal e an avois vertex e. Note that u an e can inepenently be chosen as positive or negative literals to satisfy the above conitions (A) an (B). The name quarangle is chosen because all four literals on u an e are involve in the requirement. 4. Triangle epenencies are a relaxation of Quarangle epenencies, also base on biconnecte components an articulation points of G. Specifically, tridepa( u, e ) hols uner the same conitions as quadepa( u, e ), except in conition (B) the path may start at a clause with either u or u.

11 Table 1. QBFs for Example 5.4. Ψ 1 u e t C 1 u t C 2 u t C 3 e t C 4 e t C 5 e t C 6 e t Ψ 2 u e t C 1 u t C 2 u t C 3 e t C 4 e t C 7 e t C 6 e t C 1 C 3 C 5 e C 1 C 3 C 7 e C 4 e C 4 e C 2 C 2 C 6 C 6 C 1 C 1 C 2 C 3 C 5 C 4 C 6 ee C 2 C 3 C 7 C 4 C 6 ee Ψ 1 Ψ 2 Fig. 3. (Above) Clause-literal graphs for Example 5.4. (Below) BCC-base block trees. 5. Paths for resolution-path epenencies, enote by rpdepa( u, e ), are further restricte from those for quarangle epenencies. Restrictions on paths are as follows: (C) If a path arrives at a literal noe from a clause noe, its next step must be to the complement literal. (D) If a path arrives at a literal noe from its complement literal noe, its next step must be to a clause noe. If a path goes from C 1 to literal q, then to C 2, then both C 1 an C 2 contain q. This path is allowe for triangle an quarangle epenencies, but not for resolution-path epenencies. Curiously, strict stanar epenencies relax quarangle epenencies in the opposite way from triangle epenencies. The motivation for strict stanar epenencies is that they seem to be more efficient to compute than quarangle epenencies, as iscusse later. Theorem 4.7 implies the following: Corollary 5.3 With the preceing notation: (1) If the universal variable u at qepth has no tuple (u,e) quadepa such that the qepth of e is less than + 2k, where k > 0, then u can be place at qepth + 2k in the quantifier prefix without changing the value of Ψ. (2) If existential variable e at qepth

12 + 1 has no tuple (u,e) quadepa such that the qepth of u is greater than 2k, where k > 0, then e can be place at qepth + 1 2k in the quantifier prefix without changing the value of Ψ. Example 5.4 This example illustrates resolution-path epenencies, quarangle epenencies, an their ifferences, with reference to various graph structures. Consier the close QBFs Ψ 1 an Ψ 2, given in chart form in Table 1. In the following, the notation C 1 (u) abbreviates the phrase C 1, which contains the literal u, etc., an oes not represent any operation on C 1. In both formulas a quarangle epenency quadepa( u, e ) is establishe by the paths C 1 (u) C 4 (e) an C 2 (u) C 3 (e). However, the first path is not a resolution path because oes not occur with opposite signs in C 1 an C 4. Inee, in Ψ 1 neither u nor u presses on e by any resolution path, recalling that the universal t cannot be use for connection. Therefore e is inepenent of u base on rpdepa. It follows that u an e may be exchange in the quantifier prefix without ecreasing the value of Ψ 1 (an such a swap can never increase the value). Following this exchange, it is easy to see that u may be exchange with t, then with, an universally reuce out of all clauses. Observe that Ψ 2 is the same as Ψ 1 except that it has C 7 instea of C 5. There is no obvious ifference in the chart appearance, but now C 1 (u) C 7 (e) is a resolution path an rpdepa( u, e ) hols in Ψ 2, so transposing u an e in the quantifier prefix is unsafe by this criterion. The role of the block trees is explaine in Section 6, in connection with biconnecte components an articulation points of the clause-literal graph. The efinitions are reviewe at the beginning of that section. Here we just note that the circular noe is an articulation point an the roune rectangular noes are biconnecte components. 6 Fining Depenency-Relate Paths Now we turn to the issue of computing quadepa. Biconnecte components play a central role. After reviewing the stanar theory, this section escribes how the specific information neee for quarangle epenencies is extracte. Recall that a subgraph, say B, of an unirecte graph G is biconnecte if an only if removing any one vertex an all eges incient upon that vertex oes not isconnect the remaining subgraph. A biconnecte component (BCC) of any unirecte graph G is a maximal biconnecte subgraph of G. Each ege of G is in exactly one BCC. Also, two BCCs have at most one vertex in common. A vertex that is in more than one BCC is calle an articulation point (AP). Removal of an articulation point increases the number of connecte components in G. The BCCs an APs of the epth-limite clause-literal graph G can be foun in time linear in its size. The coe in [1, Fig. 7.26] avois putting eges reunantly into the BCCs.

13 As a by-prouct, the BCC algorithm can etermine simple connecte components (CCs). An aitional by-prouct of this algorithm is the creation of an acyclic unirecte bipartite graph associate with each CC, calle the block tree, in which the BCCs are collapse to single vertices an are separate by the APs (see Figure 3). All universal literals incient upon each BCC can be collecte, as well. We continue with the terminology of Definition 5.1 for G,, u, e, etc. It is easy to etermine if there is a path in G between some clause containing u or u an a literal e in V 1 : just check if one of those clauses is in the same CC as e. Since e an e are always in the same CC, the same clauses can reach e. However, the triangle an quarangle epenency relations require paths to e an e that avoi the complement literal. If neither e nor e is an AP of G, both of these paths must exist. In this case, the relevant universal literals for e are just those that occur in some clause in the same CC as e. These sets of universal literals can be collecte once, uring the BCC algorithm. Now suppose e or e or both are APs of G. The relevant universal literals for e can be foun by starting a graph search of the block tree containing e, from e, an avoiing a visit of e. The relevant universal literals for e can be foun by starting a graph search of the block tree containing e, from e, an avoiing a visit of e. As each BCC is visite, any universal literals at qepth can be collecte. It appears that aapting this approach to compute triangle epenencies instea of quarangle epenencies will not save much time. Details are omitte for lack of space, but are straightforwar. At this time, the question of whether resolution-path epenencies can be compute in polynomial time is open. We conjecture that it is possible, but the requirement that two consecutive ege labels in the resolution graph cannot be the same makes it ifficult. 7 Empirical Data A prototype program was implemente in C++ with the Stanar Template Library to gauge the amount of variable inepenence that might be foun by various epenency relations. 2 The program computes epenency-relate quantities on QBF benchmarks. It was run on the 568 QBFEVAL-10 benchmarks. Two benchmarks ha no universal variables, so the tables inclue ata on 566 benchmarks. The platform was a 2.6 GHz 64-bit processor with 16 GB of RAM, Linux OS. The computation was limite to the outermost universal block an the ajacent enclose existential block. The number of trivial epenencies is simply the prouct of the sizes of these two blocks. The primary purpose of the program is to fin out the relative sizes of the relations for stanar epenencies, strict stanar epenencies, an quarangle epenencies. Only the outermost block pair is analyze because this provies a irect comparison between stanar 2 Please see avg/qbfeps/ for the prototype program.

14 Table 2. Eight largest QBFEVAL-10 benchmarks. Fraction of Trivial Trivial Strict Benchmark (000,000) CCs Stanar Stanar Quarangle s u-shuffle s s-shuffle s s-shuffle s s-shuffle s s-shuffle szymanski-24-s-shuffle vonneumann-rip c vonneumann-rip c Table 3. Depenency fractions as unweighte ratios. Average Avg. Fraction of Trivial Benchmark Num. in Trivial Avg. Strict Group Group (000) CCs Stanar Stanar Quarangle Eight Largest Str.St. Helpe Str.St. No Help epenencies an quarangle epenencies. Incluing multiple blocks woul obscure the size relationships because stanar epenencies use transitive closure when multiple blocks are involve, while quarangle epenencies o not. The benchmarks were partitione into several groups to try to make the statistics more informative. Table 2 shows ata for the eight largest benchmarks, as measure by the number of trivial epenencies. For six of these benchmarks, the Quarangle relation is 3-5 orers of magnitue smaller than the Trivial, while the Strict Stanar gives no reuction. On two others, no relation gives reuction. Table 3 shows the eight largest as a group, an separate the remaining benchmarks accoring to whether Strict Stanar Depenencies gave any reuction at all. Quarangle epenencies give substantial aitional reuctions, beyon stanar an strict stanar epenencies. Although Strict Stanar gave very little improvements in this test, they are essentially free, once the overhea of Stanar has been incurre. A serious question is whether the time neee to compute Quarangle Depenencies pays back in more efficient solving. Experience with epqbf inicates tentatively that Stanar Depenencies pay back in the long run [5]. For the 566 runs to get these statistics, the three longest runs took 75628, 2354, an 1561 secons. The average of the remaining 563 runs was 9.40 secons. Only fining the Strict Stanar epenencies an the BCCs average 0.50 secons on all 566 instances. Concerning the three longest runs, two of these instances have never been solve by any solver, so in a sense, nothing has been lost. However, the thir

15 instance, szymanski-24-s-shuffle, is not consiere exceptionally ifficult. It took secons to fin the quarangle epenencies, yet fining the BBCs took only three secons, an computing the Stanar Depenencies took only four aitional secons. We o not have an explanation for this outlier behavior. 8 Conclusion This paper analyzes several new epenency relations for QBF solving, an shows they form a hierarchy, together with the stanar an triangle relations propose by Samer an Szeier. The root of the hierarchy an strongest for etecting variable inepenence is the resolution-path epenency relation. Its sounness is prove; sounness of supersets (more restrictive relations) is a corollary. Whether the resolution-path relation has an efficient implementation is an open question, so quarangle epenencies, the next relation own in the lattice (Figure 1), were stuie in more etail. Computational methos for quarangle epenencies are escribe, using the theory of biconnecte components, an a prototype was implemente to gauge the sizes of BCCs an relate structures in benchmarks. Future work inclues a trial implementation of quarangle epenencies in a QBF solver, but the publicly available solvers we looke at are not goo caniates for such a retrofit by anyone except one of the original programmers, in most cases because the source coe is not public. The few with public source coe ten to lack ocumentation an contain numerous short-cuts to improve solver spee. Also, there are numerous ways to use epenencies, so one implementation experience will not be efinitive. Acknowlegment We thank Florian Lonsing an Armin Biere for many helpful iscussions. We thank the anonymous reviewers for helpful comments. References 1. Baase, S., Van Geler, A.: Computer Algorithms: Introuction to Design an Analysis. Aison-Wesley, 3r en. (2000) 2. Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantifie boolean formulas. Information an Computation 117, (1995) 3. Kleine Büning, H., Lettmann, T.: Propositional Logic: Deuction an Algorithms. Cambrige University Press (1999) 4. Klieber, W., Sapra, S., Gao, S., Clarke, E.: A non-prenex, non-clausal QBF solver with game-state learning. In: Proc. SAT, LNCS (2010) 5. Lonsing, F., Biere, A.: Integrating epenency schemes in search-base QBF solvers. In: Proc. SAT. pp Springer (2010) 6. Samer, M., Szeier, S.: Backoor sets of quantifie boolean formulas. J. Automate Reasoning 42, (2009) 7. Yates, R.A., Raphael, B., Hart, T.P.: Resolution graphs. Artificial Intelligence 1, (1970)