Dependency schemes in CDCL-based QBF solving: a proof-theoretic study

We formalize the notion of proof systems obtained by adding normal dependency schemes into the QCDCL proof system underlying algorithms for solving Quantified Boolean Formulas, by exploring the addition of the dependency schemes via two approaches: one as a preprocessing tool, and second in propagation and learnings in the QCDCL trails. We show that QCDCL augmented with the reflexive resolution path dependency scheme D rrs produces three proof systems of interest: QCDCL ( D rrs ), D rrs + QCDCL and D rrs + QCDCL ( D rrs ). We show that these three systems are not only pairwise incomparable, but also each system is incomparable with the standard QCDCL and QCDCL cube , as well as with QCDCL LEV - ORD NO - RED , Q - Res , QU - Res , and Q ( D rrs )- Res .


Introduction
Despite the NP-hardness of propositional satisfiability, SAT solvers today are amazingly efficient in solving real-world instances.The best algorithms solving SAT in practice are based on the paradigm conflict-driven clause learning CDCL, that revolutionised SAT solving in the nineties.Such algorithms use a generic template as follows: repeatedly decide values of some variables, propagate hard constraints (unit clauses) until a conflict is reached, "learn" a new clause from the conflict, backtrack and continue.For unsatisfiable formulas, the learning process yields a refutation in the proof system Resolution, and it was shown over a decade ago that resolution proofs can themselves be mimicked within this framework, so CDCL equals Resolution, [19,1].Hence, a proof-complexity-theoretic analysis of Resolution has revealed deep insights into the strengths and limitations of this CDCL paradigm.
With the success of propositional SAT solvers, there are many ambitious attempts now to tackle more expressive/succinct formalisms.In particular, for the PSPACE-complete problem of deciding the truth of Quantified Boolean Formulas QBF, there are now many solvers, as well as a rich (and still growing) theory about the underlying formal proof systems.Designing solvers for QBFs is a useful enterprise because many industrial applications seem to lend themselves more naturally to expressions involving both existential and universal quantifiers; see for instance [22,9].
The proof system Resolution can be lifted to the QBF setting in many ways.The "CDCL way" is to add a universal reduction rule, giving rise to the system Q-Res and the more general QU-Res.Allowing contradictory literals to be merged under certain conditions gives rise to the system long-distance Q-Resolution LDQ-Res.
Another "CDCL" way is to lift the CDCL algorithm itself to a QCDCL algorithm: decide values of variables, usually respecting the order of quantified alternation, propagate unit constraints, interpreting unit modulo universal reductions, repeat until a conflict is reached, learn a new clause, backtrack and continue.For false formulas, the learning process yields a long-distance Q-resolution refutation.However, the QCDCL refutation itself is much more restricted than an LDQ-Res refutation.In [7], a formal proof system QCDCL was abstracted out of the QCDCL algorithm.Noting that potentially the decision policy and the propagation policy could be modified, the authors of [7] actually formalised four different QCDCL-based proof systems.The system underlying most solvers is QCDCL LEV-ORD RED , which we will refer to as QCDCL without any sub/super-script; for the other systems we will explicitly write the policies.
While the aforementioned QCDCL proof system explains the correctness of solvers for false QBFs, it ignores cube-learning from satisfying assignments.In practice, cube-learning is essential to the completeness of a QCDCL solver; it is integral to proving a QBF to be true.The choice to ignore this was made in [7] because the focus there (as also here) was on refutational proof systems, proving QBFs false.In this setting, cube-learning is not an essential engredient.However, in [13], the authors defined the system QCDCL cube that incorporates cube-learning on top of the original QCDCL, and found that cube-learning was in fact advantageous even in constructing shorter refutations for false QBFs.
DepQBF [15] is the leading QCDCL solver and has many versions.Its base version still employs what the authors call "vanilla QCDCL", and its behaviour on false QBFs is explained by the proof system QCDCL which we are interested in exploring.Later versions of DepQBF provide options of turning on "cube learning" (when "turned on", its behaviour is explained by the proof system QCDCL cube ) and also offer heuristics like whether or not to allow "dependency scheme aware propagation" and/or apply "pure literal elimination".
A heuristic that has been found to be quite useful in many QBF solvers, and has been formalised in proof systems, is to eliminate easily-detectable spurious dependencies.In a prenex QBF, a variable "depends" on the variables preceding it in the quantifier prefix; where "depends" means that a Herbrandt/Skolem function for the variable is a function of the preceding variables.However, a Herbrand function or countermodel may not really need to know the values of all preceding variables.A dependency scheme filters out as many of such unnecessary dependencies as it can detect, producing what is in effect a Dependency QBF, DQBF.Although DQBF is a significantly richer formalism that is known to be NEXPcomplete (see [2,21]), these heuristics are not aiming to solve DQBFs in general.Rather, they algorithmically detect spurious dependencies and disregard them as the algorithm proceeds.See [14,20,15,16] for early work on this topic.Often the use of a dependency scheme makes the solvers run faster, and this is borne out by theoretical studies.Now, the universal reduction rule in the proof systems (say in Q-Res, LDQ-Res) can be applied in more settings because there are fewer dependencies, and this can shorten refutations significantly.See for instance [10,23,18].Note that the use of a dependency scheme must be proven to be sound and complete, and this in itself is often quite involved.The notion of a dependency scheme being "normal" was introduced in [18], where it is shown that adding any normal dependency scheme to LDQ-Res preserves soundness and completeness.
In this paper, we examine how the usage of a dependency scheme can affect proof systems underlying the QCDCL algorithm.As far as we are aware, such a theoretical study has not been undertaken before, even though many current QBF solvers are based on the QCDCL paradigm and also do use dependency schemes.Specifically, we focus on the proof system QCDCL (in the notation of [7], the QCDCL LEV-ORD RED proof system), underlying most QCDCL-based solvers, and on the dependency scheme D rrs which has been studied in the context of Q-Res and LDQ-Res, see [23,10,18].We note that the proof system QCDCL can be made aware of dependency schemes in more than one way.We identify two natural ways: (1) use a dependency scheme D to preprocess the formula, performing reductions in the initial clauses whenever permitted by the scheme, and (2) use a dependency scheme D in the QCDCL algorithm itself, in enabling unit propagations and in learning clauses.Denoting the first way as D + QCDCL and the second as QCDCL(D), and noting that we could even use different dependency schemes in both these ways, we obtain the system D 1 + QCDCL(D 2 ).When D 1 and D 2 are both the trivial dependency scheme D trv inherited from the linear order of the quantifier prefix, this system is exactly QCDCL.
Our contributions are as follows: 1.We formalise the proof system D ′ + QCDCL(D) for dependency schemes D, D ′ , and note that whenever D ′ , D are normal schemes, D + QCDCL(D ′ ) is sound and complete (Theorem 3.2).

2.
For D 1 , D 2 ∈ {D trv , D rrs }, we study the four systems D 1 + QCDCL(D 2 ).As observed above, one of them is QCDCL itself, while the others are new systems.We compare these systems with each other and show that they are all pairwise incomparable (Theorem 5.1).We also show that each of them is incomparable with each of the systems QCDCL LEV-ORD NO-RED , Q-Res, Q(D rrs )-Res, and QU-Res(Theorem 5.2), as well as with QCDCL cube (Theorem 5.3).In other words, making QCDCL algorithms dependency-aware is a "mixed bag": in some situations this shortens runs while in others it is disadvantageous.Here are our thoughts on what this actually means.
That QCDCL(D) is stronger than QCDCL at times is to be expected; after all, that is why the heuristic evolved.That it can be weaker at times appears a bit surprising until one recalls that even when QCDCL was formalised in [7], it was shown that the no-reduction version QCDCL LEV-ORD NO-RED can have an advantage over QCDCL; for some formulas, enabling more reductions and unit propagations can send the trails down into a trap where refuting a hard sub-formula becomes inevitable.Since dependency schemes do exactly this enabling of more reductions and propagations, custom formulas can be designed where the difference is not just between no-reductions and reductions, but also between reductions and dependency-aware reductions.This is a consequence of the level-ordering of decisions and the forcing of all unit propogations with reduction, and may not hold for the other variants of QCDCL.
That D + QCDCL can be stronger than QCDCL is again to be expected.That it can be weaker seems really counter-intuitive, but is again related to the comment above: the preprocessing shortens clauses and thus enables more unit propagations in subsequent trails.
One direction of our separation between D + QCDCL and QCDCL(D) was genuinely surprising to us.We construct formulas where after preprocessing (as in D + QCDCL) the resulting formula is propositional and easy to refute in Resolution, and hence the original formula is easy to refute in D + QCDCL.However, the same formula is hard for QCDCL(D), Section 4.5!In other words, it is not enough for the QCDCL algorithm to be dependency-aware; this awareness must be achieved at the right stage of the algorithm.
The fact that QCDCL(D), D + QCDCL, and D + QCDCL(D) are all incomparable with QCDCL cube is note-worthy and interesting as allowing for cube-learning always adds strength and makes things easier as compared to without cube-learning; QCDCL cube as a proof system is known to be strictly more powerful than QCDCL [13].Our results show that switching on cubelearning (which most current solvers do by default) and switching on dependency-awareness as proposed here are orthogonal options.Which option is better may depend on the setting from which the instances to be solved arise.
The rest of this paper is organised as follows.After spelling out notation and required prelimiaries in Section 2, including defining dependency schemes and describing the QCDCL proof system, we show in Section 3 that the addition of normal dependency schemes results in sound and complete proof systems.In Section 4 we present, for some previously studied formulas as well as for some newly designed formulas, lower and/or upper bounds in the D 1 + QCDCL(D 2 ) systems when D 1 , D 2 are in {D trv , D rrs }.Using these bounds, we conclude in Section 5 that these new systems are pairwise incomparable with each other as well as with each of QCDCL,QCDCL LEV-ORD NO-RED ,Q-Res,Q(D rrs )-Res,QU-Res,QCDCL cube .We end with some concluding remarks in Section 6.

Basics
A Quantified Boolean Formula in prenex conjunction normal form (PCNF) consists of a prefix with an ordered list of variables, each quantified either existentitally or universally, and the matrix, which is a set of clauses over these variables.That is, it has the form where φ is a propositional formula in CNF.
The formula is true if there are (Skolem) functions s i for each existentially quantified variable x i , where each such s i depends only on universally quantified variables x j with j < i, such that substituting these s i in φ yields a tautology.Similarly, the formula is false if there are (Herbrand) functions h i for each universally quantified variable x i , where each such h i depends only on existentially quantified variables x j with j < i, such that substituting h i in φ yields an unsatisfiable formula.
In this note, we focus on false formulas; refutations must rule out the existence of Skolem functions.In the proof system Q-Res, a refutation of a false QBF is a derivation of the empty clause □ from the clauses in the matrix, using two rules: Resolution (from A = C ∨ x and B = D ∨ ¬x, derive C ∨ D, provided the pivot x is existential and C ∨ D is not tautological.We denote this as C ∨ D = res(A, B, x)), and Universal Reduction (from C ∨ u derive C if u is universal and no existential variable in C appears to the right of u in the prefix).The stronger system LDQ-Res allows tautological clauses in resolution under certain conditions: a universal variable u appearing in opposite polarities in C and D is represented as the merged literal u * , provided it is to the right of the pivot x.
A proof system P is said to simulate a proof system Q if, for every formula, the size of the shortest P refutation is polynomial in the size of the shortest Q refutation.
For a set S of clauses and a literal ℓ, we use shorthand ℓ ∨ S to denote the set of clauses {ℓ ∨ C | C ∈ S}.

Dependency Schemes
Dependency schemes are mappings that associate every PCNF formula Φ with a binary relation on its variables in a manner that encodes constraints on the order of pairs of variables.The most basic of dependency schemes is the trivial dependency scheme D trv , which is in fact the order of quantifier prefix: an existential variable x depends on a universal variable u (i.e.(u, x) ∈ D trv (Φ)) if x appears to the right of u in the quantifier prefix.A non-trivial dependency scheme D produces, for any formula Φ, a subset D(Φ) of the trivial dependencies; it does not introduce new dependencies.Some non-trivial schemes are the standard scheme D std and the reflexive resolution path scheme D rrs ; see [23].Roughly speaking, in D rrs , (u, x) is in the dependency relation of a formula if (u, x) ∈ D trv and there is a sequence of clauses with the first containing u, the last containing ū, some intermediate consecutive clauses containing x and x, and where each pair of consecutive clauses has an existential variable, quantified after u, in opposite polarities.The non-existence of such a sequence implies that if at all there are Skolem functions for x, then there exists a Skolem function for x which does not use information about u; hence x need not depend on u.Formally, the dependence scheme is defined as follows: ▶ Definition 2.1 (Reflexive Resolution Path Dependency Scheme, [23]).For a QBF Φ = Qϕ, the pair (u, x) is in D rrs (Φ) if and only if (u, x) ∈ D trv (Φ) and there exists a sequence of clauses For a dependency scheme D, a QBF Φ, a universal literal ℓ u ∈ {u, ū} and an existential literal ℓ x ∈ {x, x}, we say that ℓ x blocks ℓ u if (u, x) ∈ D(Φ); in particular, this implies that x is quantified after u.For a clause C we denote by red-D(C) the subclause obtained by removing all universal literals which are not blocked by any other literal in C. We denote by red-D(Φ) the QBF Ψ obtained by replacing each clause C in the matrix of Φ with the clause red-D(C).When D = D trv , we use the notation red(C) and red(Φ).
The proof systems Q-Res and LDQ-Res, augmented with a dependency scheme D [18], permit universal reduction of u under the more relaxed requirement that (u, x) ̸ ∈ D for any existential variable x ∈ C. That is, they permit the derivation of red-D(C) from C.
An interesting and important subclass of dependency schemes are the so-called normal dependency schemes.
▶ Definition 2.2 (Normal Dependency Scheme, [18]).A dependency scheme D is monotone if for every PCNF formula ϕ, and every assignment τ to a subset of var(ϕ), D(ϕ[τ ]) ⊆ D(ϕ).(Here ϕ[τ ] is the restriction of ϕ obtained by applying the partial assignment τ to it.)simple if for every PCNF formula Φ of the form Φ = ∀XQ.ϕ,every LDQ(D)-Res derivation P from Φ, and every u ∈ X, either u or ū does not appear in P .normal if it is both monotone and simple.
If D is simple, then in a QBF with a leading universal block, universal variables from the first block appear in a LDQ-Res derivation in only one polarity.This feature is useful in proving soundness of LDQ-Res.However, for LDQ-Res(D), this alone is not enough.The additional property of monotonicity, expressing that applying a partial assignment can possibly erase existing dependencies but cannot create new ones, suffices to ensure soundness.
The dependency schemes D trv , D std , D rrs are all normal dependency schemes.These normal dependency schemes are important to us because for these dependency schemes LDQ(D)-Res is a sound proof system [18].

The proof system QCDCL
This proof system QCDCL defined in [7] formalises the reasoning in QCDCL algorithms.A refutation of a false QBF is a sequence of triples of the form (T, C, π) where T is a trail (in the QCDCL algorithm) ending in a conflict, C is the clause learnt from this trail, and π is the LDQ-Res derivation of C explaining how C is learnt.(Recall that in (Q)CDCL, a trail is a sequence of literals, some of which are decisions made by the algorithm and the others are propagated literals.Following the standard convention, we denote decision literals in a trail in boldface.)From the last triple in the sequence we can learn the empty clause, completing the refutation.Three factors affect the construction of the refutation.1.The decision policy: how to choose the next variable to branch on.In standard QCDCL (i.e QCDCL LEV-ORD RED , the focus of this paper), decisions must respect the quantifier prefix level order.(Variables x, y are at the same level if they are quantified the same way, and no variable with a different quantification appears between them in the prefix order.)Other policies such as ASS-ORD, ASS-R-ORD, UNI-ANY, are also possible; see [7,11].2. The unit propagation policy.Upon a partial assignment α to some variables, when does a clause C propagate a literal?In the No-Reduction policy, a clause C is unit if exactly one literal ℓ of C is unset, and this literal is propagated.In the Reduction policy, used by most current QCDCL solvers [15,17], a clause C propagates literal ℓ if after restricting C by α and applying all possible universal reductions, only ℓ remains.In standard QCDCL the Reduction policy is used.
In the notation of [7], for a decision policy P and a propagation policy R, the corresponding

RED
. Other variants are also defined in [7]; in particular QCDCL LEV-ORD NO-RED .3. The set of learnable clauses.These explain the conflict at the end of a trail.
▶ Definition 2.3 (learnable clauses).From a trail ending in a conflict p (r,gr) = □, the sequence L T of learnable clauses has a clause associated with each propagation in the trail, and one more clause, described by tracing the conflict backwards through the trail as follows.(ante(ℓ) denotes the clause that causes literal ℓ to be propagated; i.e. the antecedent.)C (r,gr) = red(ante(p (r,gr) )).
In the above formulation of the QCDCL system, we only consider trails that end in a conflict.Trails ending in a satisfying assignment are ignored.This is enough to ensure refutational completeness -the ability to prove all false QBFs false.From satisfying assignments, solvers can learn cubes (or terms), and this is necessary to prove true QBFs true.In [13] it was shown that allowing cube (or term) learning from satisfying assignments can also be advantageous while refuting false QBFs.This led to the definition of the proof system QCDCL cube , which was shown to be strictly stronger than the standard QCDCL system i.e.QCDCL LEV-ORD RED .Our focus, however, is on adding dependencies to the basic QCDCL system without cube learning, so wherever we talk about QCDCL as a proof system we refer to QCDCL LEV-ORD RED .

Adding Dependency Schemes to the QCDCL proof system
We first describe the generic addition of dependency schemes to QCDCL, and then show soundness and completenes for normal schemes.For a decision policy P and a propagation policy R, the corresponding QCDCL proof system is denoted QCDCL P R .Adding a dependency scheme D to this system can affect P , R, as well as the set of learnable clauses.
For the decision policy P = LEV-ORD, which is the focus of this work, adding a dependency scheme D does not affect the decision policy.
For the propagation policy, the notion of unit clauses depends on the universal reductions allowed, and this in turn is affected by the dependency scheme.In the case of R = NO-RED, no universal reductions are allowed anyway, so adding a dependency scheme to the proof system does not affect the policy.In the case of R = RED, the definition of a unit propagation changes.A clause C propagates a literal ℓ at a position in the trail if the ∀(D)-reduction of C restricted to the trail so far is a unit clause.That is, the partial assignment α specified by the trail so far does not satisfy C, and after restricting C by α, applying all universal reductions allowed by D leaves the single literal ℓ; red-D(C| α ) = {ℓ}.We denote this propagation policy as RED + D.
The dependency scheme modifies the reduction rule, which modifies the set of learnable clauses.The set of learnable clauses is now defined in a similar way as in Definition 2.3, but replacing red everywhere with red-D, the ∀(D) rule for universal reduction with respect to the dependency.
A completely different way in which a dependency scheme D can be added to QCDCL proof systems is by adding it as a preprocessing step, by applying the red-D rule on the axioms of the given formula.That is, produce QCDCL refutations of red-D(Φ) instead of Φ.
These two ways of adding dependency schemes to QCDCL -(1) in the trail construction, propagation and learning itself, or (2) as pre-processing -can both be combined.For a particular dependency scheme D, we can think of three distinct proof systems: QCDCL(D): use D for unit propagations and learning, but not for preprocessing.D + QCDCL: use D only to preprocess the formula.D + QCDCL(D): use D for preprocessing first and then use it again during propagation and learning.Going a step further, we can even use different dependency schemes in the preprocessing and in the actual trails.Thus, formally, we define the general proof system D ′ + QCDCL(D): For a false QBF Ψ = Q • ψ and a dependency scheme D, a QCDCL(D) derivation of a clause C from Ψ is a sequence of triples (T i , C i , π i ), or equivalently, a triple of sequences where for each i ∈ [m], the trail T i follows policies LEV-ORD and RED + D, each clause C j ∈ L Tj is a clause learnable from T j using the red-D rule, and If C = (□), then the derivation ι is called a refutation.
Note that QCDCL is exactly the proof system D trv + QCDCL(D trv ).Using other dependency schemes instead of D trv is a natural generalisation.
We now show that adding normal dependency schemes D 1 , D 2 preserves soundness and completness.
▶ Theorem 3.2.If D 1 and D 2 are normal dependency schemes, then D 1 + QCDCL(D 2 ) is a sound and complete proof system.
Proof.First we prove the soundness of the system.Suppose ι is a D 1 + QCDCL(D 2 ) refutation of a QBF Φ.By definition, this is a QCDCL(D 2 ) refutation of the QBF Ψ = red-D 1 (Φ).Now, every QCDCL(D) refutation has an underlying LDQ(D)-Res refutation.Therefore, from ι we can extract a LDQ(D 2 )-Res refutation Π of Ψ.Since D 2 is a normal dependency scheme, LDQ(D 2 )-Res is a sound proof system [18], and therefore Ψ is a false QBF.Now, by completeness of LDQ-Res, there exists a LDQ-Res refutation Π ′ of Ψ.The reductions made to obtain Ψ from Φ, followed by the derivation steps in Π ′ , gives a LDQ(D 1 )-Res refutation Π ′′ of Φ.Since D 1 is also a normal dependency scheme, LDQ(D 1 )-Res is also sound, and hence, the existence of Π ′′ implies that Φ is a false QBF.Now we turn to completeness.In Theorem 3.16 of the full version of [7], QCDCL (denoted there as QCDCL LEV-ORD

RED
) is shown to be complete.Exactly the same proof, which is actually quite intricate, works also to show the completeness of D 1 + QCDCL(D 2 ).The idea is as follows: for a false formula Φ, in the 2-player evaluation game, the universal player has a winning strategy on Φ.Since each clause in Φ has a subclause in Ψ = red-D 1 (Φ), the same strategy is also a winning strategy in the evaluation game on Ψ, so Ψ is false.Now, we can construct trails in level order that perform propagations whenever applicable, decide the polarity of existential variables arbitrarily, and decide the polarities of universal variables following this winning strategy.(This is possible because decisions are level-ordered.)The winning strategy guarantees that each such trail runs into a conflict.The set of learnable clauses either contains the empty clause, or is shown to contain an asserting clause -one which after backtracking becomes unit at some point in the trail -and an asserting clause is shown to be new.Thus each trail that does not terminate the refutation learns a new clause, and there are only finitely many clauses that can be added.All the arguments in this outline work also in the presence of a dependency scheme (D 2 ) that is used in both propagation and learning.◀ Having established soundness and completeness when normal dependency schemes are added, we now wish to look at how adding a particular dependency scheme affects the strength of these systems.In this work we focus on adding the reflexive resolution path dependency scheme D rrs as it is one of the most popular ones, and it is known that adding it to Q-Res gives a strictly stronger system in Q(D rrs )-Res.Therefore it is interesting to see if the same parallel extends to the QCDCL systems.Thus in the system D 1 + QCDCL(D 2 ), we will henceforth assume that D 1 , D 2 ∈ {D trv , D rrs }.When a dependency scheme is D trv , we will omit reference to it.Thus we have the systems QCDCL, QCDCL(D rrs ), D rrs + QCDCL, and D rrs + QCDCL(D rrs ).
Before proceeding further, the following propositions are noteworthy to keep in mind.If red-D rrs (Φ) is propositional, then after the preprocessing in D rrs + QCDCL and D rrs + QCDCL(D rrs ), the universal variables have no role to play and the ensuing refutation is a standard CDCL refutation.Since CDCL is equivalent to Res, the claim follows.◀ ▶ Remark 3.5.It is tempting to believe that if, for a QBF Φ, red-D(Φ) is a propositional formula easy to refute in Res, then Φ is easy to refute in QCDCL(D) as well.However, this intuition is misleading.As we will show in Section 4.5, this is provably not the case.

Refutation size bounds for some formulas
In this section we examine the effect of adding the D rrs scheme to QCDCL ( obtaining the three systems QCDCL(D rrs ), D rrs + QCDCL and D rrs + QCDCL(D rrs )) by computing bounds on refutation size for some known QBF formulas, as well as for some newly-constructed QBF formulas.

The QParity n formulas
The first family of formulas that we study are the QParity formulas, first defined in [8].
▶ Formula 1 (QParity n ).The QParity n formula has the prefix As shown in [8], these formulas are hard to refute in QU-Res (and hence also in Q-Res and QCDCL LEV-ORD NO-RED ).In [7] it was shown that they have short refutations in QCDCL.It is straightforward to see that D rrs (QParity) = D trv (QParity): the last two clauses give the dependence (z, t n ), and this extends to (z, t i ) for all i using the remaining clauses.Hence the QParity formulas are hard to refute in Q(D rrs )-Res as well.
On the other hand, since these formulas are easy to refute in QCDCL (and hence also in QCDCL cube ), from Proposition 3.3 it follows that they have short refutations in all three systems: QCDCL(D rrs ), D rrs + QCDCL, and D rrs + QCDCL(D rrs ).

The Equality n formulas
The next formula we study are another well-known family of QBF formulas, the Equality formulas, first introduced in [4].
In [4] it was shown that these formulas are hard for QU-Res, and hence also for Q-Res and QCDCL LEV-ORD NO-RED .In [7] it was shown that they are hard for QCDCL as well.However, as shown in [13], they are easy to refute in QCDCL cube .Further, as shown in [3], D rrs (Equality) = ∅, and there are short refutations in Q(D rrs )-Res.
Since D rrs (Equality) = ∅, no existential variable depends on any universal variable in the entire formula.Hence red-D rrs (Equality) is the propostional formula described below.
red-D rrs (Equality) : This formula has a short Res refutation (resolve A ′ i , B ′ i to get t i for all i, and then resolve the t i 's with T n ).Therefore by Proposition 3.4, the Equality formulas are easy to refute in D rrs + QCDCL and D rrs + QCDCL(D rrs ).
Finally we come to the system QCDCL(D rrs ).It turns out that these formulas are also easy to refute in this system, but this is not so straightforward.In particular, it does not follow merely because red-D rrs (Equality) is easy to refute in Res; see Remark 3.5.We describe the refutation below.
▶ Lemma 4.1.The Equality n formulas have O(n 2 ) refutations in QCDCL(D rrs ) Proof.Since D rrs (Equality n ) = ∅, the propagation policy and clause learning always reduce the universal u variables from the corresponding clauses.
We will construct a polynomial size refutation for the Equality n formulas containing 2(n − 1) trails.Define the following clauses: For We will construct the trails R 1 corresponding to these trails.The U trails decide x variables (as many as is possible until conflict) positively; the V trails decide them negatively.Due to the RED policy, each decision propagates at least one t literal.
The initial trail is and the antecedent clauses are ante(t j ) = B j for j ∈ [n − 1], ante( tn ) = T n , ante(x n ) = A n , and ante(□) = B n .From these set of clauses we learn the clause Restarting, create a symmetric trail to U n−1 flipping each decision: where the antecedent clauses are ante(t j ) = A j for j ∈ [n−1], ante( tn ) = T n , ante(x n ) = A n , and ante(□) = B n .From these set of clauses we learn the clause We now go down now from i = n − 2 down to i = 2.At stage i, we first construct trail U i by deciding x variables positively; we reach a conflict after deciding x i .The trail and antecedent clauses are as follows: Next, we create the symmetrical trail by deciding the x variables negatively From this we learn the clause R i .The proof ends with the two trails with antecedents ante(t 1 ) = B 1 , ante(x 2 ) = L 2 , ante(□) = R 2 , from which we learn L 1 = x1 , and finally the last trail , from which we learn the empty clause □, completing the refutation.
The QCDCL(D rrs ) refutation we have created has O(n) trails and hence overall size O(n 2 ).◀ Thus the Equality formulas, which are hard for both Q-Res and QCDCL, become easy to refute when the power of D rrs is added to these systems.Thus they showcase the power of dependency schemes and discarding spurious dependencies.

The Trapdoor n formulas
The Trapdoor formulas were introduced in [7], in order to compare QCDCL with the variant with the NO-RED policy.The idea is to juxtapose two proposotional formulas, one hard for Res and one easy for Res, and judiciously interject universal and existential variables tying the two together.The tying is done in such a way that QCDCL trails with the NO-RED policy can quickly get to the easy part, whereas with RED and the ensuing forced propagations, QCDCL is trapped into refuting the hard part.Thus for QCDCL proof systems, allowing reductions, which force more unit propagations in a trail, is not necessaririly a good thing.
▶ Formula 3 (Trapdoor n ).The Trapdoor n QBF has the prefix ∃y 1 , • • • , y sn ∀w∃t∃x 1 , • • • , x sn ∀u, where s n is the number or variables in the propositional pigeonhole principle PHP n+1 n , and the following matrix: In [7], it was shown that these formulas are easy to refute in QCDCL LEV-ORD NO-RED and Q-Res (and hence also in Q(D rrs )-Res and QU-Res), but are hard to refute in QCDCL because the reductions force unit propagations in the trails which send the solver down a "trap" of refuting PHP.
Clearly, D rrs (Trapdoor) = ∅ since the universal variables appear in only one polarity.Thus red-D rrs (Trapdoor)is the following propositional formula: This formula has a very short Res refutation involving the four y, t clauses for any i.
Hence by Proposition 3.4, the Trapdoor formulas are easy to refute in D rrs + QCDCL and D rrs + QCDCL(D rrs ).We observe below that they are quite easy to refute in QCDCL(D rrs ) as well.
Here, red-D rrs (ante(□)) = red-D rrs (y 1 ∨ w ∨ t) = y 1 ∨ t, red-D rrs (ante(t)) = red-D rrs (y 1 ∨ w ∨ t) = y 1 ∨ t and ante(ȳ 1 ) = ȳ1 Therefore, we can learn the empty clause (□), completing the refutation.◀ Thus, we see that the Trapdoor formulas which were hard for QCDCL become easy to refute when the power of D rrs is added to the QCDCL system, be it in preprocessing, unit propagation or both.This is another demonstration of the advantage of discarding spurious dependencies.

The Dep-Trap n formulas
From the previous sub-sections it may appear that adding D rrs to QCDCL only adds to its strength and suggests that the addition of the dependency scheme gives us a strictly stronger proof system as with Q-Res.However, this is not the case; QCDCL's are tricky proof systems.In the previous sections we discussed the QParity and Trapdoor formulas; these were used in [7], to show that neither of QCDCL LEV-ORD NO-RED and QCDCL simulates the other.For the Trapdoor formulas, the reduction sends the QCDCL refutation down a "trap" but not the QCDCL LEV-ORD NO-RED refutation.This motivates the idea of designing a formula where adding the dependency scheme enables reductions that send the refuatation down a trap into which the seemingly weaker systems do not fall.Based on this idea, we introduce the family Dep-Trap which is a slight modification of the Trapdoor family, and is defined as follows: ▶ Formula 4 (Dep-Trap n ).The Dep-Trap n formula has the prefix ∃y 1 , • • • , y sn ∀w∃t∀u∃x 1 , • • • , x sn , and the matrix is as given below.
Note that there are two differences from the Trapdoor n formulas.Firstly, the universal variable u which was earlier quantified at the end is now quantified just before the existential variables x 1 , • • • , x sn .Secondly, there is an additional clause w ∨ t.We will see that these serve a dual purpose: the shifting of the position of u stops QCDCL from falling into the trap, making the formulas easy to refute in QCDCL, while the additional clause prevents the D rrs scheme from bypassing the trap as it used to in the Trapdoor n formulas, since now instead of being the empty set, D rrs (Dep-Trap) = {(w, t)}.Therefore, red-D rrs (Dep-Trap) isn't a propositional formula anymore; in fact it is the following QBF: We observe below that for exactly the same reasons as Trapdoor, the Dep-Trap formulas are easy to refute in QCDCL LEV-ORD NO-RED , and hence also in Q-Res, Q(D rrs )-Res, and QU-Res.Furthermore, the shifting of u to before the x i 's stops QCDCL from going down the "trap", and hence the formulas are easy to refute in QCDCL (and in QCDCL cube ).Proof.First we will show that the Dep-Trap formulas have a linear-size refutation in QCDCL.We will do so by constructing a complete refutation.The first trail decides all y variables positively, decides w negatively, and then propagates t and a conflict.Unlike Trapdoor we don't propagate an x after an y decision, since the u cannot be reduced and so blocks unit propagation.
Since none of the unit propagations in the two trails above employed a universal reduction, the same refutation is also a valid linear-size refutation in QCDCL LEV-ORD NO-RED .Since all the other systems mentioned in the lemma simulate QCDCL Proof.D rrs (Dep-Trap n ) = {(w, t)}.This means that the universal variable w cannot be reduced from any axiom clause it appears in as they all also contain the variable t, whereas the universal variable u can be reduced from the axiom clauses as no existential variable depends on it.Let us first see hardness for QCDCL(D rrs ).Since u can be reduced but not w, the proof of hardness of Trapdoor in QCDCL from [7] carries over as is to hardness of Dep-Trap in QCDCL(D rrs ): the decisions on the y variables propagate x literals, sending the trails down the PHP trap.The additional clause cannot change the course of such trails, because its variables appear after the y part in the prefix and decisions are required to be level-ordered.
Next consider the other two variants.Preprocessing yields the QBF red-D rrs (Dep-Trap) described above.Now all trails in a refutation of this formula must start with deciding the y i 's; these decisions propagate the x i 's; and w cannot be reduced from the clauses it exists in (irrespective of whether we allowed ∀(D)-reductions or not) nor decided before all the y i 's are decided.Again, the proof of hardness of Trapdoor in QCDCL from [7] lifts to hardness of Dep-Trap in D rrs + QCDCL and D rrs + QCDCL(D rrs ).◀ The Dep-Trap formulas are thus easy for QCDCL, but become hard to refute when D rrs is added to the system demonstrating that allowing more reductions and removing spurious dependencies does not necessarily help for the QCDCL system.

The TwoPHPandCT n formulas
The formulas in the previous sections seem to suggest that Proposition 3.4 could also extend to include QCDCL(D rrs ).We show now that this is not the case.The motivation for defining the following formula also comes from the Trapdoor formulas, using the propositional hardness of PHP and the "easiness" of the (negation of) complete tautology on two variables.The added new element is the use of two disjoint copies of the hard part.
▶ Formula 5 (TwoPHPandCT n ).The TwoPHPandCT n formulas has the prefix, Observe that these formulas are easy to refute in Q-Res, using the four z 1 , z 2 clauses, and hence also easy to refute in Q(D rrs )-Res and QU-Res.
Since v appears in only one polarity and u, ū appear in clauses with disjoint variables, hence D rrs (TwoPHPandCT) = ∅ and red-D rrs (TwoPHPandCT) is the propositional formula This formula is easy to refute in Res using the z 1 , z 2 clauses; hence by Proposition 3.4, the original QBFs are easy to refute in D rrs + QCDCL and D rrs + QCDCL(D rrs ).
The final question arises now as to how hard they are to refute in QCDCL, QCDCL(D rrs ) and QCDCL LEV-ORD NO-RED .And the answer is that it is hard for all three systems.
Proof.None of the three systems QCDCL, QCDCL(D rrs )or QCDCL LEV-ORD NO-RED allow for any preprocessing.Hence the first decision in each of these three systems must be on u, which allows for no propagations in any case.And depending on the choice of u, the next set of decisions are either all on x variables or all on y variables.The variable v could be dropped during unit propagation in the QCDCL(D rrs ) system, but neither z 1 or z 2 could be decided or propagated before all the y or x variables are decided/propagated.Therefore, all conflicts these trails hit come directly from the PHP clauses.Thus refuting TwoPHPandCT in QCDCL, QCDCL(D rrs ) or QCDCL LEV-ORD NO-RED is equivalent to refuting PHP in CDCL, requiring exponential size.◀ These formulas highlight two important facts: firstly that QCDCL(D rrs ) is not the same as D rrs + QCDCL or D rrs + QCDCL(D rrs ) and does not simulate them either.And secondly, even in the case when D rrs = ∅ and reducing the formula by D rrs gives us an easy propositional formula, it can still be hard to refute for QCDCL(D rrs ).

The RRSTrapEq n formulas
The next family of formulas are obtained by making a slight modification to the Equality formulas.The motivation to define such a formulas comes from trying to ascertain whether after preprocessing with D rrs , does allowing reductions using D rrs for unit propagation add any power over trivial universal reductions.
▶ Formula 6 (RRSTrapEq n ).The RRSTrapEq n formula has the prefix ∃b and the PCNF matrix as below: The prefix has the variables of Equality sandwiched between a and b.The underlying idea in the formulation is that unlike D rrs (Equality) which is empty, the additional Further, adding b to the x, u, t clauses along with that results in red-D rrs (RRSTrapEq) = RRSTrapEq.Now, since preprocessing by D rrs does not change the formula at all, therefore the refutational hardness and in fact the refutation for these two formulas will be exactly the same for the pair of systems QCDCL and D rrs + QCDCL, and similarly for the pair QCDCL(D rrs ) and D rrs + QCDCL(D rrs ).
▶ Lemma 4.6.The RRSTrapEq formulas have polynomial size refutations in QCDCL(D rrs ) and D rrs + QCDCL(D rrs ), but require exponential size refutations in QCDCL and D rrs + QCDCL Proof.Since, red-D rrs (RRSTrapEq)= RRSTrapEq, there is no effect of preprocessing, and all 4 systems start off by refuting the same formula.Now, for any of the four proof systems, if you consider any trail T of a refutation, the first decision must be on the variable a, and irrespective of the manner of that decision we propagate b, which satisifies all the C i clauses and removes the b literal from the A i and B i clauses.The remaining formula at this point is exactly the Equality formula, i.e.RRSTrapEq| a= * ,b=0 = Equality, and hence refuting the RRSTrapEq formulas is equivalent to refuting the Equality formulas in QCDCL and QCDCL(D rrs ), which we know are hard and easy respectively.
Therefore, RRSTrapEq formulas have polynomial size refutations in QCDCL(D rrs ) and D rrs + QCDCL(D rrs ), but require exponential size refutations in QCDCL and D rrs + QCDCL ◀ Additionally, since the Equality formulas are embedded in the RRSTrapEq formula, and since QU-Res is closed under restrictions, the RRSTrapEq formulas are hard for QU-Res and in turn Q-Res and QCDCL LEV-ORD NO-RED .However, they are easily seen to have short refutations in Q(D rrs )-Res: resolve the last two clauses to derive b, use it to remove b from the A i and B i , now use D rrs reductions to remove the u literals, resolve on x to derive unit t i clauses, and remove them from T n sequentially.

The PreDepTrap n formulas
The previous section underlined that preprocessing by D rrs may not necessarily make D rrs during propagation obsolete and/or give an advantage.It is reasonable to believe that at least it won't make things worse.But the following example shows that this is not the case: preprocessing before allowing a QCDCL system to refute could in fact make the refutation harder.
The construction of the formula is pretty straightforward.It is the disjoint union of the two formulas Dep-Trap and Equality, connected by a universal quantified right at the beginning, appearing in the two sub-formulas in opposite polarites.
▶ Formula 7 (PreDepTrap n ).The PreDepTrap n formula has the prefix ∀a ∃y First consider the QCDCL LEV-ORD NO-RED and QCDCL systems.Since there is no preprocessing, the first decision of every trail must set the variable a.For a trail that starts with the decision ā, the PreDepTrap formula reduces exactly to Dep-Trap formula which we know is easy to refute in QCDCL LEV-ORD NO-RED as well as QCDCL (Section 4.4).Therefore, the PreDepTrap formulas are easy to refute in QCDCL and QCDCL LEV-ORD NO-RED , and as a consequence in Q-Res, Q(D rrs )-Res, QU-Res, and QCDCL cube .
Next, consider adding D rrs .Since the opposite polarities of a appear in clauses with disjoint non-interacting sets of variables, no existential variable depends on a.Since D rrs (Equality) = ∅, no variable depends on a q variable.Thus D rrs (PreDepTrap) = D rrs (Dep-Trap), In the QCDCL(D rrs ) system, where there is no preprocessing, the first decision of every trail must again set the variable a.For trails that start with the decison a, the formula immediately reduces exactly to the Equality formulas, which are easy to refute in QCDCL(D rrs ), Lemma 4.1.The refutation in the proof of Lemma 4.1, preceded by the decision a, gives a valid refutation for the PreDepTrap formulas.Therefore these formulas are easy to refute in QCDCL(D rrs ).
Finally, consider the case when the formula is preprocessed.We show that this makes refutations exponentially long.
Now the variable a, though still in the quantifier prefix, has no effect on the trails.It can be decided arbitrarily in the beginning, and then the goal is to refute red-D rrs (PreDepTrap) in QCDCL and QCDCL(D rrs ).At this point, therefore, all trails must start with deciding the y i 's, propagating the x i 's.Since the only remaining universal w cannot be reduced from the clauses it occurs in nor decided before all the y i 's are decided, and since the p i 's or r i 's also cannot be decided before the y i 's, the trails are led into the trap of refuting PHP.The proof of hardness of Trapdoor in QCDCL from [7] carries over exactly as it is to show hardness for D rrs + QCDCL and D rrs + QCDCL(D rrs ).◀

The PropDep-Trap n formulas
The previous section illustrated an example where having D rrs as a preprocessing technique was a detriment to refuting it and in fact it was better to use D rrs only in unit propagation, and not also for preprocessing.This leads to the question -could there be a formula where having D rrs only for preprocessing was strictly better than having it for both preprocessing and propagation?Addressing this led to the birth of the following formula which is a slight modification of the Dep-Trap formulas, and which witnesses that the answer is yes.
▶ Formula 8 (PropDep-Trap n ).The PropDep-Trap n formulas have the prefix , z 2 and the matrix as given below.
First observe that the presence of the four s, w, t clauses make this formula very easy to refute in Q-Res and hence in Q(D rrs )-Res and QU-Res.
Next, notice that the formulas are also easy to refute in QCDCL LEV-ORD NO-RED , QCDCL, and QCDCL cube because, after the initial propagation of the unit clauses z1 , z2 in level 0, there are no propagations possible before a w decision; the decisions on s and all the y i 's cause no propagations.A trail that decides s and w quickly reaches a conflict and learns s; a next trail that propagates s and decides w then learns the empty clause.) refutation, so it suffices to show hardness for the latter.Let us observe how a QCDCL(D rrs ) refutation looks.The two unit clauses z1 , z2 get propagated initially and then s and the y i 's have to be decided.A decision on s, irrespective of the polarity, causes no further propagations until w is also decided, as every clause containing the variable s also contains the variables w and t, and the reduction of w is blocked by t.
Since neither (b 1 , x i ) nor (b 2 , x i ) is in D rrs , a decision on variable y i propagates either x i (due to the clause containing ȳi , where b 1 can now be reduced), or xi (due to the clause containing y i , where b 2 can now be reduced).The propagating of x i due to y i sends the QCDCL(D rrs ) refutation down the same "trap" as the Trapdoor formulas for QCDCL [7], and thus QCDCL(D rrs ) requires exponential size refutations to refute these formulas.◀

The TwinEq n formulas
The TwinEq formulas were introduced in [13] to show hardness in the system QCDCL cube .They are formally defined as follows: ▶ Formula 9 (TwinEq n [13]).The TwinEq n formula has the prefix These formulas are hard for QCDCL cube , and hence also for QCDCL and QCDCL LEV-ORD NO-RED .For the same reason as for Equality (the size-cost-capacity theorem from [4]), they are also hard for QU-Res and Q-Res.
It is easy to show that D rrs (TwinEq) = ∅ (just as D rrs (Equality) is shown to be ∅).Therefore, red-D rrs (TwinEq) is a propositional formula, and in fact is the same formula as red-D rrs (Equality).Therefore, due to the same argument as for the Equality formulas in Section 4.2, the TwinEq formulas are easy to refute in Q(D rrs )-Res, D rrs + QCDCL, D rrs + QCDCL(D rrs ) and QCDCL(D rrs ).

Relation between proof systems
The previous section saw us study the bounds for known as well as our newly-constructed formulas in a plethora of proof systems.Using these obtained bounds, in this section we look to obtain the relations of our newly defined proof systems using D rrs with QCDCL among themselves, as well as their relation with other QBF proof systems.First we observe that the four versions of QCDCL that use or do not use D rrs in either of the two ways are all pairwise incomparable.The Dep-Trap formulas (Section 4.4) have constant size refutations in P 2 but require exponential size refutations in P 1 .
◀ Finally, we observe that even when we add cube-learning to standard QCDCL, the system QCDCL cube is still incomparable with all the three versions of QCDCL with dependency scheme added.

Conclusion
We have examined, from a rigourous proof-theoretic viewpoint, the effect of incorporating heuristics based on dependency schemes into QBF solving algorithms based on the conflictdriven clause learning paradigm.Our results show that unlike in the case of the proof system Q-Res, where dependency-awareness can shorten but never lengthens refutations, here the picture is much more nuanced, and all kinds of shortenings as well as lengthenings can be observed.Thus the decision of whether or not to make a QCDCL solver account for spurious dependencies is itself a challenging one, and it is likely the domain from where instances are to be solved may indicate what choice is more suitable.One aspect which is unresolved in our work is the nature of our upper bounds.Since QCDCL is not in itself an algorithm but a template, there are multiple instantiations of it based on the choice of decision heuristics, propagation policy, and learning scheme.We have restricted ourselves here to the decision heuristic and propagation policy used in most state-of-the-art solvers, namely, level-ordered decisions and propagations with reductions.However, we have not specified the learning scheme.Our lower bounds hold for any QCDCLbased solver as long as the learning scheme picks a clause only from the learnable-clause sequence as defined in Section 2. However, the upper bounds hold for specific choices of learnt clauses, and this, in some sense, reflects a certain non-determinism in the algorithm.(This is somewhat akin to the non-determinism inherent in CDCL algorithms in the statement that CDCL simulates Resolution.)Arguably, the upper bounds will be more meaningful if achieved with actually-used learning schemes.While some of our upper bounds are achieved with such schemes, specifically the UIP policy, some others make ad hoc choices with respect to which clauses to learn.Thus the impact of the learning scheme itself is still improperly understood for dependency-aware QCDCL.
In this work, we only consider decisions in LEV-ORD.Whether incorporating dependency schemes in the decision policy itself makes a difference or not is an interesting question.Formalisation seems tricky as it seems to be more like D-ORD than LEV-ORD(D).As recent work in [12] shows, other orders are not necessarily unsound, so this can still be meaningful.However, this is quite a different direction to explore.There is also the dependency-learning setup which is quite different: the solver starts off assuming there are no dependencies, and gradually builds up a set of dependencies.(That is, add required dependencies starting from ∅, rather than remove spurious dependencies starting from D trv .)In this approach, explored in [17] and used in the QBF solver Qute, learning dependencies does affect decision order.An in-depth future comparison of these two approaches could be interesting to explore.
Another aspect we have not directly addressed, other than showing soundness and completeness, is the effect of dependency schemes other than D rrs .Since D rrs is what is actually used in most QBF solvers, it is important to understand its effect first.But there are other schemes: the less (than D rrs ) general standard dependency scheme D std introduced in [20], and more general schemes based on tautology-free and implication-free paths introduced in [5,6].It seems reasonable to expect that a similar nuanced picture will present when considering such schemes as well.
Finally, an aspect to our work that could also be interesting to investigate, with potential ramifications to practical solvers, is exploring the addition of cube-learning to our dependency-aware QCDCL systems.This may be somewhat non-trivial and nuanced, as adding dependency schemes to the formal proof system of long-distance term resolution (the proof system in which proofs can be extracted from runs of solvers on true QBFs, Q-consensus) is not known to be sound (see the Discussion section in [18]).

▶ Proposition 3 . 4 .
then all of QCDCL, QCDCL(D), D + QCDCL, and D + QCDCL(D)are equivalent on Φ and produce the same refutations.This is simply because if D = D trv , then adding the dependency scheme gives nothing new to the system as no extra reductions are enabled.For a QBF Φ, D rrs (Φ) = ∅ if and only if red-D rrs (Φ) is a propositional formula (no universal variables in any clause).Further, if D rrs (Φ) = ∅, then red-D rrs (Φ) is easy to refute in Res if and only if Φ is easy to refute in D rrs + QCDCL and D rrs + QCDCL(D rrs ).Proof.(Sketch) If D rrs (Φ) = ∅, then by definition red-D rrs (Φ) is propositional.If D rrs (Φ) ̸ = ∅, then there is some reflexive resolution path involving a universal variable u, and the occurrence of u in the first clause of the path is blocked by an existential literal even with respect to D rrs .So red-D rrs (Φ) is not propositional.

▶ Lemma 4 . 4 .
LEV-ORD NO-RED , Dep-Trap has polynomial-size refutations in these proof systems as well.◀ Next we show that these formulas are hard to refute in QCDCL variants that use D rrs .Refutations of the Dep-Trap n formulas in QCDCL(D rrs ), D rrs + QCDCL and D rrs + QCDCL(D rrs ) require exponential size.
Trap is the disjunction of a with every clause of Dep-Trap, and similarly for ā ∨ Equality.)

▶ Theorem 5 . 1 .▶ Theorem 5 . 2 .
The proof systems in {QCDCL, QCDCL(D rrs ), D rrs + QCDCL, D rrs + QCDCL(D rrs )} are pairwise incomparable.Proof.Of the four systems under consideration, the Trapdoor formulas (Section 4.3) are hard only for QCDCL, and the Dep-Trap formulas (Section 4.4)are easy only in QCDCL.Hence QCDCL is incomparable with all three systems obtained by adding D rrs .Among the three systems using D rrs , the TwoPHPandCT formulas (Section 4.5) are hard only in QCDCL(D rrs ), while the PreDepTrap formulas (Section 4.7) are easy only in QCDCL(D rrs ).Hence QCDCL(D rrs ) is incomparable with the systems that use preprocessing.Finally, the systems D rrs + QCDCL and D rrs + QCDCL(D rrs ) are separated by the formulas RRSTrapEq (Section 4.6) easy only in D rrs + QCDCL(D rrs ), and the formulas PropDep-Trap (Section 4.8) easy only in D rrs + QCDCL.◀Next we observe that each of the three new versions of QCDCL is also incomparable withQCDCL LEV-ORD NO-RED , Q-Res, Q(D rrs )-Res, and QU-Res.Any two proof systems P 1 ∈ {QCDCL(D rrs ), D rrs + QCDCL, D rrs + QCDCL(D rrs )}, and P 2 ∈ {QCDCL LEV-ORD NO-RED , Q-Res, Q(D rrs )-Res, QU-Res}, are incomparable.Proof.The QParity formulas (Section 4.1) require exponential size refutations in P 2 but have polynomial size refutations in P 1 .