Learning-Sensitive Backdoors with Restarts
- Edward Zulkoski
1.University of WaterlooWaterlooCanada
2.Carnegie Mellon UniversityPittsburghUSA
3.Microsoft ResearchCambridgeUK
4.University of TorontoTorontoCanada
Zulkoski E. et al. (2018) Learning-Sensitive Backdoors with Restarts. In: Hooker J. (eds) Principles and Practice of Constraint Programming. CP 2018. Lecture Notes in Computer Science, vol 11008. Springer, Cham. https://doi.org/10.1007/978-3-319-98334-9_30
Abstract
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Restarts are a pivotal aspect of conflict-driven clause-learning (CDCL) SAT solvers, yet it remains unclear when they are favorable in practice, and whether they offer additional power in theory.译文：重新启动是冲突驱动的clauso -learning (CDCL) SAT解决程序的一个关键方面，但它们在实践中什么时候是有利的，以及它们在理论上是否提供了额外的能力仍然不清楚。 In this paper, we consider the power of restarts through the lens of backdoors. 译文：我们从后门的角度来考虑重启的力量。Extending the notion of learning-sensitive (LS) backdoors, we define a new parameter called learning-sensitive with restarts (LSR) backdoors. 译文：扩展学习敏感(LS)后门的概念，我们定义了一个新的参数，称为学习敏感的重启(LSR)后门。Broadly speaking, we show that LSR backdoors are a powerful parametric lens through which to understand the impact of restarts on SAT solver performance, and specifically on the kinds of proofs constructed by SAT solvers.译文：广义地说，我们表明LSR后门是一个强大的参数透镜，通过它可以理解重启对SAT求解器性能的影响，特别是对SAT求解器构造的证明类型的影响
First, we prove that when backjumping is disallowed, LSR backdoors can be exponentially smaller than LS backdoors.译文：首先，我们证明了当反向跳被禁止时，LSR后门可以比LS后门成倍地小。 Second, we demonstrate that the size of LSR backdoors are dependent on the learning scheme used during search.译文：其次，我们证明了LSR后门的大小依赖于搜索过程中使用的学习方案。 Finally, we present new algorithms to compute upper-bounds on LSR backdoors that intrinsically rely upon restarts, and can be computed with a single run of a CDCL SAT solver.译文：最后，我们提出了新的算法来计算LSR后门的上界，本质上依赖于重新启动，并且可以通过CDCL SAT求解器的一次运行来计算。
We empirically demonstrate that this can often produce much smaller backdoors than previous approaches to computing LS backdoors.译文：我们的经验证明，这通常可以产生比以前计算LS后门方法小得多的后门。 We conclude with empirical results on industrial benchmarks which demonstrate that rapid restart policies tend to produce more “local” proofs than other heuristics, in terms of the number of unique variables found in learned clauses of the proof.译文：我们以工业基准上的经验结果得出结论，这些结果表明，根据在证明的学习子句中发现的唯一变量的数量，快速重启策略往往比其他启发式方法产生更多的“局部”证明。 |
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Keywords
SAT solving Backdoors Restarts CDCL
1 Introduction
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Restarts are a pivotal aspect of conflict-driven clause-learning (CDCL) SAT solvers. Not only are they crucially important to the performance of CDCL solver implementations, but foundational theoretical work on the power of CDCL intrinsically relies on exploiting restarts [1, 24]. For example, the powerful theorems by Pipatsrisawat and Darwiche that show polynomial equivalence between CDCL SAT solvers (with perfect branching and restarts) and the general resolution proof system seem to crucially rely on the use of restarts. 译文：Pipatsrisawat和Darwiche的强大定理表明CDCL SAT解算器(具有完美的分支和重新启动)和一般的解算证明系统之间的多项式等价性，似乎关键依赖于重新启动的使用。
Nevertheless, it remains unclear when or why it is favorable to restart in practice, and whether restarts truly make CDCL SAT solvers a more powerful proof system in theory. 译文：，在实践中何时以及为什么重启是有利的，重启是否真正使CDCL SAT求解器在理论上成为一个更强大的证明系统还不清楚。 Further, it is unclear whether restarts provide any additional power over backjumping, which is standard in all modern CDCL SAT solvers. 译文：此外，还不清楚重启是否提供了比backjump更强大的能力，后者是所有现代CDCL SAT解决程序的标准。 Understanding restarts can go a long way in explaining one of the most important problem in SAT and SMT solver research, namely, “why are solvers efficient for a large class of industrial applications?”译文：理解重启有助于解释SAT和SMT求解器研究中最重要的问题之一，即“为什么求解器对大类别的工业应用是有效的?” |
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Complexity theorists have long proposed a variety of parameters in explaining the surprising power of certain algorithms at solving NP-complete problems. One such class of parameters, originally introduced by Williams, Gomes, and Selman, are backdoors [29].译文：复杂性理论家长期以来一直提出各种参数来解释某些算法在解决np完全问题方面的惊人能力。最初由威廉姆斯、戈麦斯和塞尔曼引入的一类参数是后门[29]。 Since their seminal paper, a variety of backdoors have been proposed in an attempt to understand why and for what kind of instances SAT solving algorithms work efficiently.译文：自从他们的重要论文发表以来，各种各样的后门被提出，试图理解为什么和为什么类型的实例，求解算法有效工作。 Intuitively, backdoors measure how many variables need to be assigned such that a polynomial-time subsolver can solve the residual formula.译文：直观上，后门测量了需要分配多少变量，以便多项式时间子求解器可以求解残差公式。 For “traditional” types of backdoors (i.e. strong and weak backdoors [29]), if the backdoor is small, then efficient algorithms can determine satisfiability by trying all possible assignments to the backdoor.译文：对于“传统”类型的后门(即强后门和弱后门[29])，如果后门很小，那么有效的算法可以通过尝试所有可能的后门分配来确定满足性。 Unfortunately, traditional backdoors do not account for some pivotal aspects of CDCL SAT solvers, such as clause-learning or restarts.译文：不幸的是，传统的后门没有考虑到CDCL SAT解决程序的一些关键方面，如clauseclearn或重启。 Dilkina et al. extended traditional backdoors to learning-sensitive (LS) backdoors in order to account for the power of clause-learning during the search performed by a SAT solver [9].译文：Dilkina等人将传统的后门扩展为学习敏感(LS)后门，以说明在SAT求解器[9]执行搜索时，集群学习的威力。 They showed that these learning-sensitive (LS) backdoors are exponentially smaller than traditional strong backdoors on certain class of formulas. However, they only considered a single configuration of a CDCL SAT solver, namely one that uses the first unique implication point (1UIP) learning scheme and disallows restarts.译文：但是，他们只考虑了CDCL SAT求解器的单一配置，即使用第一个惟一暗示点(1UIP)学习方案并不允许重启的配置。 |
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| In this work, we extend the notion of LS backdoors to allow restarts by introducing the concept of learning-sensitive with restarts (LSR) backdoors. 译文：我们通过引入restart (LSR)后门的学习敏感概念来扩展LS后门的概念，从而允许重新启动。Our main contribution is an exponential separation between LS and LSR backdoors (using the 1UIP clause-learning scheme), under the condition that backjumping during search is not allowed (only backtracking is allowed). 译文：我们的主要贡献是LS和LSR后门之间的指数分离(使用1UIP clause-learning scheme)，条件是不允许在搜索期间回溯(只允许回溯)。Determining whether or not restarts add significant power to CDCL SAT solvers in full generality remains a major and important open problem. We hope that our work will be a useful step toward tackling this problem.译文：确定重新启动是否在完全通用的情况下为CDCL SAT解决程序增加了显著的能力仍然是一个主要的、重要的开放问题。我们希望我们的工作将是解决这个问题的有益的一步。 | |
| We further consider the effect of different clause-learning schemes on the size of LSR backdoors. 译文：我们进一步考虑了不同的clausellearning方案对LSR后门规模的影响。We show that if a formula has a backdoor set B under the decision learning (DL) scheme (an “LSR-DL backdoor”), then B is also an LSR-1UIP backdoor. The converse however does not hold, and we show a family of formulas where the smallest LSR-1UIP backdoor is exponentially smaller than the smallest LSR-DL backdoor. This may indicate that 1UIP can allow the solver to remain more “local” during search when compared to other heuristics. | |
| In the context of strong and weak backdoors, if we are given a priori knowledge of a backdoor B of size n, then we can simply invoke the subsolver on all 2n2n assignments to the backdoor to determine satisfiability. The situation is not so simple in the context of LSR (and LS) backdoors, which rely upon the order in which the search space is explored to learn clauses. We demonstrate pitfalls that may arise for a solver trying to exploit a priori knowledge of LSR backdoors. We describe formulas such that if the solver is given additional unit clauses for free (thus shrinking the search space), then it is impossible to determine unsatisfiability by only branching on the LSR backdoor (of the original formula without the unit clauses). We also describe issues that can arise if multiple conflicts can be learned after the same sequence of decisions. Under certain probabilistic assumptions, we show that even if the solver is given a perfect branching sequence witnessing the backdoor, it still may have to branch on additional variables. This result exploits the fact that a typical CDCL solver only learns one clause per conflict. | |
| Finally, through an extension of results from [24], we show that the set of variables found in the learned clauses used in the proof of unsatisfiability constitute an LSR backdoor. We use this result to empirically compare the proofs produced by various restart policies. Results suggest that rapid restart policies tend to have more “local proofs” (in terms of the aforementioned measure) on several classes of industrial SAT formulas. | |
2 Related Work
| Williams et al. introduced traditional weak and strong backdoors for both SAT and CSP [29]. The size of backdoors with respect to subsolvers different from unit propagation (UP) was considered in [10, 17]. Several extensions of traditional strong and weak backdoors have been proposed. Backdoor keys measure certain interdependencies of variables in the backdoor set [26]. Backdoor trees refine strong backdoors by measuring what fraction of the total search of a strong backdoor must be explored before determining satisfiability [27]. The backdoor treewidth is a measure that is bounded above by both the strong backdoor size and also the treewidth of certain graphical abstractions of the formula [11]. It was shown in [16] that there is little relationship between weak backdoors and the backbone of a formula. Dilkina et al. were the first to consider clause learning in the context of backdoors and introduced learning-sensitive (LS) backdoors. LS backdoors allow clause-learning to occur while traversing the assignment tree of backdoor variables, which yield exponentially smaller backdoors than strong backdoors for certain class of instances [8, 9]. | |
| Our work is inspired by several lines of work aimed at relating the power of CDCL to general resolution. Pool resolution was first introduced to model CDCL without restarts [28], and it was shown that pool resolution is exponentially stronger than regular resolution. Resolution trees with lemmas were similarly introduced in [7], and more closely match clause-learning algorithms in practice. In their seminal paper, Beame et al. formally defined CDCL as a proof system and showed that CDCL can polynomially simulate natural refinements of general resolution [2]. However, their approach required assumptions that do not reflect typical CDCL implementations, such as choosing to ignore unit propagations when preferable. Hertel et al. also showed that CDCL without restarts can effectively polynomially simulate general resolution, but required certain modifications to input formulas [14]. It was also shown that a non-restarting CDCL solver can polynomially simulate a restarting solver, but the approach requires adding additional variables to the formula as a “counter,” based on the number of variables in the original formula [4]. | |
| Recent approaches that show CDCL solving efficiently simulates general resolution require restarts. In their papers [24, 25], the authors showed that CDCL without the assumptions from [3] can polynomially simulate general resolution. The approach relies upon the notion of 1-empowerment [23], which is the dual of clause absorption [1]. However, crucially, they assume that the branching and restarts in CDCL solvers are perfect (i.e., non-deterministic). In Atserias et al. [1], the authors assume randomized branching and restarts, instead of non-deterministic ones. Specifically, they demonstrate that rapidly restarting solver with sufficiently many random decisions can effectively simulate bounded-width resolution. Many questions in this context remain open. For example, can the above simulations be modified to not require restarts? Further, can we construct realistic models of CDCL solvers (with “realistic” branching and restarts) and determine their relative power vis-a-vis well-known proof systems such as general resolution. | |
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On the empirical side, several works have studied the performance of various restart policies. Huang reported on a comprehensive evaluation of several restart policies [15,2007], which demonstrated the strength of “dynamic” restart policies such as those based on the Luby sequence [19，1993]. Biere performed an evaluation of restart strategies on more modern solvers [5,2015]. Among their results, they showed that static restart policies can perform as well as dynamic strategies.译文：在他们的结果中，他们表明静态重启策略的性能可以和动态重启策略一样好。 Haim et al. showed that more rapid restart policies tend to require fewer conflicts before determining satisfiability, however this does not always lead to faster solving [13,2013].译文：Haim等人的研究表明，在确定可满足性之前，更快的重启策略往往需要更少的冲突，但这并不总是能够更快地解决[13]。
——重启的最新文献这么少吗？ |
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3 Background
5.2 Empirical Results |
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| We compare several solving and restart heuristics through the lens of this spanning variables metric, which we will refer to simply as the LSR backdoor of the proof (LSR in Tables 1 and 2). Our experiments are conducted over all unsatisfiable instances from both the Application track of the SAT competition from 2009–2014, as well as the Agile 2016 instances. The Agile track contains smaller instances generated from the quantifier-free bit-vector formulas derived from the SAGE whitebox fuzzer [12]. The timeout in the SAT competition for these instances was only 60 s. | |
| We consider three restart policies: (1) the Luby heuristic; (2) restarting after every conflict (“Always”); and (3) never restarting. For the Agile instances, we considered three branching heuristics: LRB [18], VSIDS [21], and random branching (with phase-saving polarity selection), thus totaling 9 solver configurations in combination with the restart policies. For the Application instances, we did not include VSIDS or random branching in our experiments due to the cost of computation and to avoid the random branching heuristic greatly limiting our set of usable instances. We allotted 10,000 s for each Application instance, and 300 s for each Agile instance. Experiments were run on an Azure cluster, where each node contained two 3.1 GHz processors and 14 GB of RAM. Each experiment was limited to 6 GB. We only include instances where we could compute data for all heuristics being considered, in total, 1168 Agile instances, and 81 Application instances. For each instance, the size of the LSR backdoor is normalized by the total number of variables. | |
| Tables 1 and 2 depict the results. On average, the always-restart policy seems to produce more local proofs than the other policies, regardless of the branching heuristic. This may provide further explanation as to why restarts are useful in practice, particularly on unsatisfiable instances. Interestingly, the always-restart policy ends up requiring the most time and conflicts to solve the Application instances; this may indicate that the usefulness of this locality is dependent on the types of instances. We also wish to emphasize that although the average LSR ratio is only 0.03 smaller for always-restart than the other policies on Application instances, this amounts to approximately 390 variables on average. | |
| Finally, we compare our above approach to computing LSR backdoors to the previously proposed “All Decisions” approach to computing LS backdoors. In [9], the authors compute LS backdoors by running a randomized non-restarting CDCL solver to completion and recording the set of all variables branched upon during search. This set constitutes an LS backdoor. The process is repeated many times to try to find small backdoors. Due to the number of instances we consider, we only use one run of the solver for each heuristic being considered. Tables 1 and 2 compare our above LSR approach to the set of all decision variables (computed on the same solver run with restarts). Since many clauses learned during search are not useful for the proof, the all-decisions approach records many unnecessary decisions that are ultimately not useful. The result does not hold on many types of crafted instances however, particularly when the formula is designed to intrinsically require proofs spanning many variables. Nonetheless, our approach seems to work for certain classes of instances found in industrial settings. | |
5 Relating LSR Backdoors to CDCL Proofs
| We next show connections to LSR backdoors and the proofs generated by CDCL solvers on unsatisfiable instances. Specifically, we show that if B is the union of all variables found in the “useful clauses” of the proof, then B is an LSR backdoor for the formula. More importantly, we show that frequent restarts often result in smaller and more “local” proofs. | |
6 Conclusions and Future Work
| In this paper, we explored several important questions within the context of the broad research program of trying to understand why Boolean SAT solvers are so efficient for industrial instances obtained from verification, program analysis, testing, and synthesis. 译文：在这篇论文中，我们在试图理解为什么布尔SAT求解器对于从验证、程序分析、测试和综合中获得的工业实例如此有效的广泛研究计划中探索了几个重要的问题。All the questions we explored relate to restart techniques in SAT solvers, new kinds of restart-aware backdoors we introduced, and perhaps most importantly a characterization of the properties (e.g., locality) of proofs produced by SAT solvers through the lens of such backdoors.译文：我们探索的所有问题都与SAT求解器中的重启技术有关，我们引入了新的可感知重启的后门类型，也许最重要的是，通过这种后门的透镜，SAT求解器证明的特性(例如，局部性)的描述。 | |
| Specifically, we introduced the notion of LSR backdoors, and demonstrated an exponential separation from LS backdoors (which in turn were shown to be exponentially smaller than strong backdoors for certain class of instances in previous work by Dilkina et al.). A takeaway of this result is that clause learning together with restarts is capable of exploring the search space in ways not possible with clause learning alone.译文：我们引入了LSR后门的概念，并证明了与LS后门的指数分离(在Dilkina等人之前的工作中，对于某些实例类，LS后门的指数比强后门小)。这个结果的一个结论是子句学习与重新启动一起能够以单独子句学习不可能的方式探索搜索空间。 | |
| We further showed that LSR-1UIP backdoors may be exponentially smaller than LSR-DL backdoors. The order in which the search space is explored is crucial when branching over both LS and LSR backdoors, and we demonstrated several issues that may arise during the search. Empirically, we demonstrated that rapid restart strategies tend to produce significantly more local proofs than other strategies on industrial instances.译文：我们进一步表明，LSR-1UIP后门可能比LSR-DL后门小成指数。当在LS和LSR后门上分支时，搜索空间的搜索顺序是至关重要的，我们演示了在搜索过程中可能出现的几个问题。在实证上，我们证明了快速重启策略比其他策略在工业实例上产生更多的局部证据。 | |
| Going forward, we would like to refine our empirical results further by comparing uniform restart policies (e.g., restarting every k conflicts or geometric restarts) that are less “extreme” against the always-restart-at-conflicts policy. We also plan to refine our notion of locality in proofs by considering the structure of (e.g., the variable-incidence graph) of Boolean formulas. Another line of future work is to answer the big open question, namely, are CDCL SAT solvers with restarts a more powerful proof system than CDCL without restarts?译文：接下来，我们将通过比较不那么“极端”的统一重启策略(例如，重新启动每一个k冲突或几何重启)与总是在冲突中重新启动策略，进一步完善我们的实证结果。我们还计划通过考虑布尔公式的结构(如变关联图)来改进证明中的局部性概念。未来工作的另一个方向是回答一个大的开放问题，即，有重启的CDCL SAT求解器是否比没有重启的CDCL更强大的证明系统? | |
References
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