文献学习-Guiding CDCL SAT Search via Random Exploration amid Conflict Depression

Guiding CDCL SAT Search via Random Exploration amid Conflict Depression

Md Solimul Chowdhury, Martin M¨ uller, Jia-Huai You
Department of Computing Science, University of Alberta
Edmonton, Alberta, Canada.
{mdsolimu, mmueller, jyou}@ualberta.ca

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Abstract

The efficiency of Conflict Driven Clause Learning (CDCL) SAT solving depends crucially on finding conflicts at a fast rate. State-of-the-art CDCL branching heuristics such as VSIDS, CHB and LRB conform to this goal.译文：冲突驱动子句学习(CDCL) SAT求解的效率关键取决于快速发现冲突的速度。最先进的CDCL分支启发法，如VSIDS、CHB和LRB符合这一目标。   We take a closer look at the way in which conflicts are generated over the course of a CDCL SAT search. Our study of the VSIDS branching heuristic shows that conflicts are typically generated in short bursts, followed by what we call a conflict depression phase in which the search fails to generate any conflicts in a span of decisions.译文：我们将进一步了解在CDCL SAT搜索过程中产生冲突的方式。我们对VSIDS分支启发式的研究表明，冲突通常在短时间内产生，然后是我们所说的冲突压抑阶段，在这个阶段，搜索在一系列决策中无法产生任何冲突。 The lack of conflict indicates that the variables that are currently ranked highest by the branching heuristic fail to generate conflicts.译文：缺乏冲突表明，目前排名最高的分支启发式变量不能产生冲突。   Based on this analysis, we propose an exploration strategy, called expSAT, which randomly samples variable selection sequences in order to learn an updated heuristic from the generated conflicts. The goal is to escape from conflict depressions expeditiously.译文：在此基础上，我们提出了一种名为expSAT的探索策略，该策略对变量选择序列进行随机采样，以便从生成的冲突中学习更新的启发式算法。目标是迅速摆脱冲突萧条。   The branching heuristic deployed in expSAT combines these updates with the standard VSIDS activity scores. An extensive empirical evaluation with four state-of-the-art CDCL SAT solvers demonstrates good-to-strong performance gains with the expSAT approach.译文：expSAT中部署的分支启发式将这些更新与标准的VSIDS活动分数结合起来。对四个最先进的CDCL SAT解题器进行了广泛的实证评估，结果表明expSAT方法的性能提升很好，甚至很强。

Introduction

	Modern CDCL SAT solvers have become the enabling technology for many real-world problems, such as hardware design verification (Gupta, Ganai, and Wang 2006), classical planning (Rintanen 2012) and encryption (Massacci and Marraro 2000). The key decision-making step in a CDCL SAT solver is selecting a variable from the current set of unassigned variables using a branching heuristic, before making a boolean assignment to it. Variable selection has a dramatic effect on search efficiency. Popular branching heuristics include VSIDS (Moskewicz et al. 2001) and its variants, LRB (Liang et al. 2016b) and CHB (Liang et al. 2016a). These heuristics reward variables involved in recent conflicts. The intuition is that assignments of these variables are likely to generate further conflicts, leading to useful learned clauses and thus pruning the search space.
	Global Learning Rate (GLR) (Liang et al. 2017) measures the number of conflicts obtained per branching decision. In CDCL, a single decision may generate multiple conflicts. State-of-the-art branching heuristics, such as LRB, VSIDS or CHB, have average GLR values of about 0.5, i.e., average one conflict per two decisions (Liang et al. 2017).
	In this work, we first perform a study of conflict generation during CDCL search. We find that there are clear nonrandom patterns of short bursts of conflicts, called conflict burst (CB), followed by longer phases of what we call conflict depression (CD), in which the search fails to generate any conflicts in a span of decisions.译文：在本文中，我们首先对CDCL搜索过程中的冲突生成进行了研究。我们发现有明显的非随机模式的短期冲突爆发，称为冲突爆发(CB)，然后是较长的阶段，我们称为冲突抑郁(CD)，在这个阶段中，搜索在一系列决策中没有产生任何冲突。 To correct the course of such a search, we propose to use exploration to combat conflict depression. We therefore design a new SAT solver extension, called expSAT, which applies random walks in the context of CDCL SAT solving.译文：为了纠正这种探索的过程，我们建议使用探索来对抗冲突抑郁症。因此，我们设计了一个新的SAT求解器扩展，称为expSAT，它将随机游动应用于CDCL SAT求解。 In a conflict depression phase, random walks help to identify more promising variables for branching. Note that exploration visits possible future search states, while the standard CDCL branching heuristics rely on conflicts generated from past search states.译文：在冲突抑郁阶段，随机行走有助于识别更有希望的分支变量。注意，探索访问可能的未来搜索状态，而标准的CDCL分支启发式依赖于从过去搜索状态生成的冲突。
	The contributions of this paper are:
	An empirical study of the pathological state of conflict depression, using one of the strongest purely VSIDS-based solvers, glucoseLCM1 (gLCM), on recent SAT competition benchmarks, shows that CD phases occur at a high rate and often with long average duration. 译文：在最近的SAT竞赛基准上，一项关于冲突抑郁病理状态的实证研究，使用最强的纯vsid解决方法之一glucoseLCM1 (gLCM)，显示CD期发生的频率很高，而且通常平均持续时间很长。
	A formulation of expSAT for an exploration-driven extension of VSIDS-based SAT solvers. expSAT performs random exploration when a substantial CD phase is detected.The goal is a swift escape from conflict depression. expSAT的一种配方，用于基于vsid的SAT求解器的勘探驱动扩展。当检测到大量CD期时，expSAT执行随机探索。目标是迅速逃离冲突压抑。
	An empirical evaluation of expSAT implemented on top of four state-of-the-art SAT solvers, gLCM, MapleCOMSPS (MplCOMSPS), Maple CM (MplCM) and MapleLCMDist ChronoBT (MplCBT). On the benchmarks of main track of SAT Competitions 2017 and 2018 (SAT-2017 and SAT-2018), all four expSAT extensions solve more instances and achieve lower PAR-2 score2 than their respective baselines.译文：在2017年和2018年的SAT竞赛(SAT-2017和SAT-2018)的基准上，所有四种扩展都解决了更多的实例，并实现了低于各自基线的PAR-2得分。 The best performing expSAT solver solves 16 more instances than its baseline, which is a strong performance gain.译文：性能最好的expSAT求解器比它的基线多解决16个实例，这是一个强大的性能增益。 On 52 hard instances from SATCoin cryptographic benchmarks, most of our expSAT extensions show strong gains over their respective baselines.译文：在SATCoin加密基准测试的52个硬实例中，我们的大多数expSAT扩展都比它们各自的基线获得了强劲的增益。
	An analysis of the experimental results shows that our results are consistent with two standard performance metrics, GLR and average LBD (Liang et al. 2017). In addition, exploration reduces average length of CD phases.
	An algorithm to update exploration parameters during the search and an experimental comparison of this adaptive version of expSAT with the non-adaptive one3.
	Propagation Depression Amid a CD Phase During a CD phase, VSIDS scores are not a good predictor of a variable’s future performance, since branching decisions fail to produce any conflict and perform only truth value propagations.译文：在CD阶段，VSIDS分数并不能很好地预测变量未来的性能，因为分支决策不会产生任何冲突，只执行真值传播。 Are there any differences in the pattern of unit propagations between CD and CB phases? We define the Propagation Rate (PR) as the number of propagations per decision. Table 1 compares the average PR values for Test Set 1 over the decisions in CD and CB phases. On average, PR values during a CD phase are almost 10 times lower than CB phases. Clearly, this result demonstrates that during a CD phase, VSIDS branching decisions go through propagation depression as well.译文：显然，这个结果表明，在CD阶段，VSIDS的分支决策也会经历传播抑制。

 

Preliminaries

 

Conflict Depression and Conflict Bursts

	Consider a run of a CDCL SAT solver Ψ which makes a total of d decisions. In each decision, a variable is selected according to a branching heuristic. Each decision i (0 < i ≤d) leads to some number ci ≥ 0 of conflicts. Let us represent the conflict history of the search by the sequence of ci and define a conflict depression (CD) phase as a sequence of one or more consecutive decisions with no conflict. Let us define a conflict burst (CB) phase as a sequence of one or more consecutive decisions with at least one conflict. Let us further define the length of a CD and CB phase as the number of decisions in it E.g., the conflict history of decisions (1,0,0,0,0,4,2,1,0,1,0,0), where a number represents the number of conflicts at that decision, contains 3 CD phases: one starting at decision 2 with length 4, one starting at decision 9 with length 1, and one at the end with length 2. It also contains 3 CB phases at decision 1 with length 1, at decision 6 with length 3 and at decision 9 with length 1.
	DR, CDR, FDC, FDOC and FDMC: Suppose the solver Ψ takes a total of d decisions, encounters u CD phases and gives r restarts. We define the Decision Rate (DR) as d/r and CD phase Rate (CDR) as u/r. We also define Fraction of Decisions with Conflicts (FDC) as the measure of the fraction of decisions which produce at least one conflict. This measure is related to but different from GLR. It counts decision with conflicts, not conflicts. We further partition FDC into FDOC+FDMC, where  FDOC and FDMC are Fraction of Decisions with One Conflict (FDOC) and Fraction of Decisions with Multiple Conflicts (FDMC), respectively.

 

Conflict Depression in gLCM

	Conflict Depression in gLCM
	We now study conflict depression empirically, using VSIDS as a representative CDCL branching heuristic, and gLCM as the underlying SAT solver. We collect the statistics for each search of an instance on DR, CDR, Average CD phase length, GLR, FDC, FDOC and FDMC.
	CDR and Average CD Phase Length       The left plot of Fig.1 shows the Decision Rates (DR), CD phase Rate (CDR) and average CD phase length (in log scale) for instances in Test Set 1, where the instances are sorted by average CD phase length. We observe that the average CD phase length is short for most instances, but still consists of multiple decisions (blue). Furthermore, irrespective of their average CD phase length, for all-most all instances CD phases (orange) occur at a high rate given the decision rates (yellow). The histogram on the right side of Fig. 1 shows the distribution of average length of CD phases. This average ranges from 2.09 to 1402.30. 263 instances have a very short length (at most 3). The distribution is heavy-tailed, with 69 instances of average length greater than 25 (rightmost bin). Overall, the data indicates that for gLCM on Test Set 1, conflict depressions occur frequently and often last over multiple decisions (high average CD phase length).
	Propagation Depression Amid a CD Phase During a CD phase, VSIDS scores are not a good predictor of a variable’s future performance, since branching decisions fail to produce any conflict and perform only truth value propagations. Are there any differences in the pattern of unit propagations between CD and CB phases?   We define the Propagation Rate (PR) as the number of propagations per decision.译文：我们定义传播速率(PR)为每个决策的传播次数。 Table 1 compares the average PR values for Test Set 1 over the decisions in CD and CB phases. On average, PR values during a CD phase are almost 10 times lower than CB phases. Clearly, this result demonstrates that during a CD phase, VSIDS branching decisions go through propagation depression as well.译文：平均而言，CD阶段的PR值几乎比CB阶段低10倍。显然，这个结果表明，在CD阶段，VSIDS的分支决策也会经历传播抑制。