Limits of CDCL Learning via Merge Resolution

A common feature of CDCL solvers is the use of 1-empowering learning schemes [PD08, AFT11]: that is, they only learn clauses which enable unit propagations that were not possible before. An example of 1-empowering learning scheme is the popular \lfstyle1UIP learning scheme [MS99]. To model this behavior we build upon a connection between 1-empowerment, and merges [And68], i.e., resolution steps involving clauses with shared literals.

Nearly every CDCL solver nowadays uses the First Unique Implication Point (\lfstyle1UIP) learning scheme, where conflict analysis starts with a clause falsified by the current state of the solver and sequentially resolves it with clauses responsible for unit propagations leading to the conflict, until the clause becomes asserting, i.e., unit immediately upon backjumping.

Descriptions of early implementations of CDCL solvers [MS99, MMZ⁺01] already remark on the importance of learning an asserting clause, since that nudges the solver towards another part of the search space, and consequently early alternative learning schemes explored learning many kinds of asserting clauses. First observe that conflict analysis can be extended to produce other asserting clauses that appear after the \lfstyle1UIP during conflict analysis such as intermediate UIPs and the last UIP [BS97]. The early solver GRASP can even learn multiple UIP clauses from a single conflict. While there is empirical evidence that it is often best to stop conflict analysis at the \lfstyle1UIP [ZMMM01], recent work has identified conditions where it is advantageous to continue past it [FB20] (see also the discussion of learning schemes therein).

Ryan [Rya04, §2.5] also observed empirically that clause quality is negatively correlated with the length of the conflict analysis derivation and considered the opposite approach, that is, learning clauses that appear before the \lfstyle1UIP during conflict analysis in addition to the \lfstyle1UIP. This approach is claimed to be useful for some empirical benchmarks but, like any scheme that learns multiple clauses, slows down Boolean constraint propagation (BCP) in comparison to a scheme that learns just the \lfstyle1UIP.

Later works provide a more theoretically oriented approach to understanding the strength of \lfstyle1UIP and to learning clauses that appear before the \lfstyle1UIP [DHN07, PD08]. In particular, and highly relevant for our discussion, Pipatsrisawat and Darwiche identified 1-empowerment as a fundamental property of asserting clauses. Furthermore they identified a connection between 1-empowering clauses and merges, and used the simplicity of checking for merges as an approximation for 1-empowerment.

An orthogonal approach is to extend the \lfstyle1UIP derivation by resolving it with clauses other than those that would usually be used during conflict analysis [ABH⁺08]. A prominent example is clause minimization [SB09], where literals are eliminated from the \lfstyle1UIP clause by resolving it with the appropriate input clauses, independently of their role in the conflict, so the resultant clause that is actually learned is a shorter and therefore stronger version of the \lfstyle1UIP.

Furthermore, a relation between merges and unit-resolution completeness has also been observed in the context of knowledge compilation [dV94]. Finally, the amount of merges directly inferable from a formula (i.e., in a single resolution step) has been proposed, under the name of mergeability, as a measure to help explain the hardness of a formula based on both controlled experiments as well as analysis of real-world instances [ZMW⁺18].

To summarize, merges are relevant in the context of CDCL learning schemes for the following reason: all practical CDCL learning schemes either produce a 1-empowering clause or extend one, and since 1-empowering clauses always contain a merge in its derivation, we have that all practical learning schemes produce a clause that contains a merge in its derivation, which is exactly the property imposed by the proof systems we introduce below.

As mentioned earlier, we build upon a connection between 1-empowerment and merges [PD08, AFT11], and introduce a proof system RMA (for “resolution with merge ancestors”) which includes CDCL with an arbitrary 1-empowering learning scheme. The “merge ancestors” in the name of this system comes from the fact that for any 1-empowering clause, at least one step in its resolution derivation must resolve two clauses that share a common literal: a merge step in the sense of [And68]. Clause minimization procedures, as long as they are applied on top of 1-empowering clauses, are also modelled by RMA.

We prove that, on the one hand, RMA is able to simulate resolution only with a linear overhead. On the other hand, we show a quadratic separation between resolution and RMA, that is there exist formulas with resolution proofs of linear length that require RMA proofs of quadratic length. That is, we show that CDCL may be polynomially worse than resolution because of the properties of a standard learning scheme, but that the blow-up due to these properties is not more than linear.

We also consider weaker proof systems, all of which contain \lfstyle1UIP (and do so with finer granularity), but not necessarily other asserting learning schemes. A technical point of interest is that we work with proof systems that are provably not closed under restrictions, which is unusual in proof complexity. This fact forces our proof to exploit syntactic properties of the proof system, as opposed to relying on more convenient semantic properties.

We need to define a few concepts from CDCL proofs. An in-depth treatment can be found in the Handbook of Satisfiability [BN21]. Fix a CNF $F$, also known as clause database. A trail $\tau$ is a sequence of tuples $(x_{j_{i}}=b,C_{i})$ where $C_{i}$ is either a clause in $F$ or the special symbol $d$ representing a decision. We denote by $\alpha_{<i}$ the assignment $\{x_{j_{i}}=b\mid i^{\prime}<i\}$, and we denote by $\mathrm{dl}(i)=\mathrm{dl}(i-1)+\llbracket{C_{i}=d}\rrbracket$ the decision level at position $i$, that is the number of decisions up to $i$. We mark the position of the last decision in a trail by $i^{*}$.

A trail is valid if for every position $i$ that is not a decision we have that ${C_{i}}{\upharpoonright_{\alpha_{<i}}}=x_{j_{i}}^{b}$ and for every decision $i$ we have that for every clause $C\in F$ such that ${C}{\upharpoonright_{\alpha_{<i}}}=x^{b}$, the literal $x^{b}$ appears in the trail before $i$. In particular, for every position $i^{\prime}<i$ with $\mathrm{dl}(i^{\prime})<\mathrm{dl}(i)$ we have ${C_{i}}{\upharpoonright_{\alpha_{<i^{\prime}}}}\neq x_{j_{i}}^{b}$.

A clause $C$ is asserting if it is unit at the last decision in the trail, that is ${C}{\upharpoonright_{\alpha_{<i^{*}}}}=x^{b}$. It is 1-empowering if $C$ is implied by $F$ and can lead to new unit propagations after being added to $F$, that is if there exists a literal $\ell\in C$ such that for some $A\in\{\bot,\ell\}$, it holds that $F\land\overline{C\setminus\ell}\nvdash_{1}A$. If a clause is not 1-empowering then we say it is absorbed by $F$.

Given a clause $D_{\lvert\tau\rvert}$ falsified by a trail $\tau$, the conflict derivation is an input derivation $D_{\lvert\tau\rvert},\ldots,D_{k+1},D_{k}$ where $D_{i-1}=\mathrm{Res}(D_{i},C_{i})$ if $x_{j_{i}}\in D_{i}$, and $D_{i-1}=D_{i}$ otherwise. The first (i.e., with the largest index) asserting clause in the derivation is called the \lfstyle1UIP. Note that $D_{i^{*}}$ is always asserting (because $D_{i}$ is falsified by $\alpha_{\leq i}$ for $i^{*}\leq i\leq\lvert\tau\rvert$ and $D_{i^{*}}$ is not falsified by $\alpha_{<i^{*}}$), therefore we can assume that the \lfstyle1UIP always has index at least $i^{*}$.

We call a sequence of input derivations input-structured if the last clause of each derivation can be used as an axiom in successive derivations. The last clause of each but the last derivation is called a lemma. A CDCL derivation is an input-structured sequence of conflict derivations, where learned clauses are lemmas. This definition is similar to that of Resolution Trees with Input Lemmas [BHJ08], with the difference that the sequence only needs to be ordered, without imposing any further tree-structure on the global proof.

The following Lemmas highlight the practical relevance of merges by relating them to \lfstyle1UIP, asserting, and 1-empowering clauses.

Let $\ell,m,n$ be positive integers. We have variables $x_{i}$ for $i\in[m\ell-1]$ and $w_{j,k}$ for $j\in[\ell]$ and $k\in[n]$. For convenience we define $x_{0}=1$ and $x_{m\ell}=0$, which are not variables. Let $X=\{x_{i}\mid i\in[m\ell-1]\}$, $W_{j}=\{w_{j,k}\mid k\in[n]\}$ and $W=\bigcup_{j\in[\ell]}W_{j}$. For each $j\in[\ell]$ we build the following gadget:

Each equality is expanded into the two clauses $B_{j,k,1}=w_{j,k}\lor\overline{w_{j,k+1}}$ and $B_{j,k,0}=\overline{w_{j,k}}\lor w_{j,k+1}$, and we collectively call them $\mathcal{W}=\{B_{j,k,b}\mid j\in[\ell],k\in[n-1],b\in\{0,1\}\}$. Observe that the $j$-th gadget implies $w_{j,1}=w_{j,n}$. Additionally we build the following gadget:

where $\hat{\imath}\in[\ell]$ denotes the canonical form of $i\pmod{\ell}$. Each constraint is expanded into the two clauses $A_{i,1}=w_{\hat{\imath},1}\lor w_{\hat{\imath},n}\lor\overline{x_{i-1}}\lor x_{i}$ and $A_{i,0}=\overline{w_{\hat{\imath},1}}\lor\overline{w_{\hat{\imath},n}}\lor\overline{x_{i-1}}\lor x_{i}$, and we collectively call them $\mathcal{X}=\{A_{i,b}\mid i\in[m\ell],b\in\{0,1\}\}$. The resulting formula is called $F_{\ell,m,n}$.

It is not hard to see that there is a resolution refutation of $F_{\ell,m,n}$ of length $\mathrm{O}(\ell\cdot(m+n))$. Indeed, we first derive the two clauses representing $w_{j,1}=w_{j,n}$ for each $j\in[\ell]$, which requires $\mathrm{O}(n\ell)$ steps:

Then we resolve each of the $\mathcal{X}$ axioms with one of these clauses, appropriately chosen so that we obtain pairs of clauses of the form $w_{\hat{\imath}}^{b}\lor\overline{x_{i-1}}\lor x_{i}$ for $i\in[m\ell]$, and resolve each pair to obtain the chain of implications $x_{1},\ldots,x_{i}\rightarrow x_{i+1},\ldots,\overline{x_{n\ell-1}}$ in $\mathrm{O}(m\ell)$ steps.

Since we have derived a chain of implications $x_{1}$, $x_{1}\rightarrow x_{2}$, …, $x_{m\ell-1}\rightarrow x_{m\ell-1}$, $\overline{x_{m\ell-1}}$ we can complete the refutation in $\mathrm{O}(m\ell)$ more steps. Let us record our discussion.

Before we prove the lower bound let us discuss informally what are the natural ways to refute this formula in RML, so that we understand which behaviours we need to rule out.

If we try to reproduce the previous resolution refutation, since we cannot reuse the clauses representing $w_{j,1}=w_{j,n}$ because they are not merges, we have to rederive them each time we need them, which means that it takes $\mathrm{O}(mn\ell)$ steps to derive the chain of implications $x_{1},\ldots,x_{i}\rightarrow x_{i+1},\ldots,\overline{x_{n\ell-1}}$. We call this approach refutation 1. This refutation has merges (over $w_{\hat{\imath},1}$, $x_{i-1}$, and $x_{i}$) when we produce $w_{\hat{\imath},1}^{b}\lor\overline{x_{i-1}}\lor x_{i}$, and (over $x_{i-1}$ and $x_{i}$) when we produce $\overline{x_{i-1}}\lor x_{i}$, but since we never reuse these clauses the refutation is in fact tree-like.

An alternative approach, which we call refutation 2, is to start working with the $\mathcal{X}$ axioms instead. In this proof we clump together all of the repeated constraints of the form $w_{j,1}\neq w_{j,n}$ for every $j\in[\ell]$, and then resolve them out in one go. In other words, we first derive the sequence of constraints

where $D_{i}$ can be obtained from $D_{i-1}$ and the pair of $\mathcal{X}$ axioms $A_{i,b}$, then resolve away the inequalities from $D_{m\ell}=\bigvee_{j\in[\ell]}w_{j,1}\neq w_{j,n}$ using the $\mathcal{W}$ axioms. However, representing any of the constraints $D_{i}$ for $i\geq\ell$ requires $2^{\ell}$ clauses, which is significantly larger than $mn\ell$ and even superpolynomial for large enough $\ell$, so this refutation is not efficient either. Note that this refutation has merges (over $W$ variables) each time that we derive $D_{i}$ with $i\geq\ell$.

A third and somewhat contrived way to build a refutation is to derive the pair of clauses representing $w_{j,1}=w_{j,n}$ using a derivation whose last step is a merge, so that they can be reused. Each of these clauses can be derived individually in $\mathrm{O}(mn\ell)$ steps, for a total of $\mathrm{O}(mn\ell^{2})$ steps, by slightly adapting refutation 5.2, substituting each derivation of $x_{i}\rightarrow x_{i+1}$ by a derivation of $w_{j,1}\lor\overline{w_{j,n}}\lor\overline{x_{i}}\lor x_{i+1}$ whenever $i\equiv j\pmod{\ell}$ so that at the end we obtain $w_{j,1}\lor\overline{w_{j,n}}$ instead of the empty clause. Such a substitution clause can be obtained, e.g., by resolving $w_{j,1}\lor w_{j,2}\lor\overline{x_{i}}\lor x_{i+1}$ with $\overline{w_{j,2}}\lor\overline{w_{j,n}}\lor\overline{x_{i}}\lor x_{i+1}$ as follows

After deriving $w_{j,1}=w_{j,n}$ as merges we follow the next steps of refutation 5.2 and complete the refutation in $\mathrm{O}(m\ell)$ steps. We call this refutation 3.

Observe that the minimum length of deriving the clauses representing $w_{j,1}=w_{j,n}$ is only $\mathrm{O}(n)$, even in RML, so if we only used the information that refutation 5.2 contains these clauses we would only be able to bound its length by $\Omega(\ell\cdot(m+n))$. Therefore when we compute the hardness of deriving a clause we need to take into account not only its semantics but how it was obtained syntactically.

Before we begin proving our lower bound in earnest we make two useful observations.

Our first step towards proving the lower bound is to rule out that refutations like refutation 5.2 can be small, and to do that we show that wide clauses allow for very little progress. This is a common theme in proof complexity, and the standard tool is to apply a random restriction to a short refutation in order to obtain a narrow refutation. However, RMA is not closed under restrictions, as we prove later in Corollary 5.12, and because of this we need to argue separately about which merges are preserved.

Let us define the class of restrictions that we use and which need to respect the structure of the formula. A restriction is an autarky [MS85] with respect to a set of clauses $D$ if it satisfies every clause that it touches; in other words for every clause $C\in D$ either ${C}{\upharpoonright_{\rho}}=1$ or ${C}{\upharpoonright_{\rho}}=C$. A restriction is $k$-respecting if it is an autarky with respect to $\mathcal{W}$ axioms, we have ${F_{\ell,m,n}}{\upharpoonright_{\rho}}\cong F_{k,m,n}$ up to variable renaming, and every $X$ variable is mapped to an $X$ variable. Our definition of a narrow clause is also tailored to the formula at hand, and counts the number of different $W$-blocks that a clause $C$ mentions. Formally $\mu(C)=\lvert\{j\in[\ell]\mid\exists x_{j,k}\in\mathrm{vars(C)}\}\rvert$.

Note that $s({\pi}{\upharpoonright_{\rho}})$ is a resolution refutation of ${F_{n,\ell}}{\upharpoonright_{\rho}}$, but not necessarily a RMA refutation, therefore we lose control over which clauses may be reused²²2Recall that $s(\pi)$ is the syntactic equivalent of $\pi$.. Nevertheless, we can identify a fragment of $s({\pi}{\upharpoonright_{\rho}})$ where we still have enough information.

Note that $C_{t}$ may be semantically implied by the $\mathcal{W}$ axioms, and have a short derivation as in refutation 5.2, therefore we are forced to use syntactic arguments to argue that deriving $C_{t}$ using an $\mathcal{X}$ axiom takes many resolution steps.

The next step is to break $\psi$ into $m$ (possibly intersecting) parts, each corresponding roughly to the part of $\psi$ that uses $\mathcal{X}$ axioms with variables in an interval of length $\ell$ (by Lemma 5.4 we can assume that $\psi$ contains axioms from every interval). To do this we use the following family of restrictions defined for $i\in[n]$:

Let $X_{i}=X\cap\sigma_{i}^{-1}(*)$ and note that ${F_{\ell,m,n}}{\upharpoonright_{\sigma_{i}}}\cong F_{\ell,1,n}$.

Clauses in $\psi$ with many $X$ variables could be tricky to classify, but intuitively it should be enough to look at the smallest positive literal and the largest negative literal, since these are the hardest to eliminate. Therefore we define $r(C)$ to be the following operation on a clause: literals over $W$ variables are left untouched, all positive $X$ literals but the smallest are removed, and all negative $X$ literals but the largest are removed. Formally,

where $x_{\min A}$ (resp. $\overline{x_{\max B}}$) is omitted if $A$ (resp. $B$) is empty.

We need the following property of $r(C)$.

We define each part $\psi_{i}$ to consist of all clauses $C\in\sigma$ such that $C$ is

1. 1.

   an $\mathcal{X}$ axiom not satisfied by $\sigma_{i}$; or

2. 2.

   the conclusion of an inference with pivot in $X_{i}$; or

3. 3.

   the conclusion of an inference with pivot in $W$ that depends on an $\mathcal{X}$ axiom if $r(C)$ contains a variable in $X_{i}$; or

4. 4.

   the conclusion of an inference with pivot in $W$ that does not depend on $\mathcal{X}$ axioms if the only immediate successor of $C$ is in $\psi_{i}$.

This is the point in the proof where we use crucially that the original derivation is in RMA form: because clauses that do not depend on $\mathcal{X}$ axioms are not merges, they have only one successor and the definition is well-formed.

Ideally we would like to argue that parts $\psi_{i}$ are pairwise disjoint. This is not quite true, but nevertheless they do not overlap too much.

To conclude the proof we need to argue that each $\psi_{i}$ is large. The intuitive reason is that $\psi_{i}$ must use one $\mathcal{X}$ axiom for each $j\in[(i\ell,(i+1)\ell]$, which introduces a pair of $W$ variables from each $W_{j}$ block, but since no clause contains more than $\ell/8$ such variables, we need to use enough $\mathcal{W}$ axioms to remove the aforementioned $W$ variables. Formally the claim follows from these two lemmas.