文献学习——Making Deduction More Effective in SAT Solvers

Making Deduction More Effective in SAT Solvers

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 29, NO. 8, AUGUST 2010

	Abstract Satisfiability (SAT) solvers often benefit from transformations of the formula to be decided that allow them to do more through deduction and decrease their reliance on enumeration. For formulae in conjunctive normal form, subsumed clauses may be removed or partial resolution may be applied. The objectives of simplifying the formula and speeding up the solver are sometimes competing. We characterize existing transformations in terms of their impact on the deductive power of the formula and their effects on the sizes of the implication graphs. For example, we show that variable elimination works by improving implication graphs. We also present two new techniques that try to increase deductive power. The first is a check performed during the computation of resolvents. The second is a new preprocessing algorithm based on distillation that combines simplification and increase of deductive power. Most current SAT solvers apply resolution at various stages to derive new clauses or simplify existing ones. The former happens during conflict analysis, while the latter is usually done during preprocessing. We show how subsumption of the operands by the resolvent can be inexpensively detected during resolution; we then discuss how this detection is used to improve three stages of the SAT solver: variable elimination, clause distillation, and conflict analysis. The “on-the-fly” subsumption check is easily integrated in a SAT solver. In particular, it is compatible with strong conflict analysis and the generation of unsatisfiability proofs. Experiments show the effectiveness of the new techniques. Index Terms—CNF, distillation, DPLL, preprocessing, SAT.   可满足性(SAT)求解器通常受益于要确定的公式的转换，这允许他们通过演绎来做更多的事情，并减少对枚举的依赖。 对于合取范式的公式，可以去掉包含子句或采用部分分解。简化公式和加快求解速度的目标有时是相互矛盾的。 我们根据它们对公式的演绎能力的影响以及它们对蕴涵图大小的影响来描述现有的变换。例如，我们通过改进隐含图来证明变量消除是有效的。我们还提出了两种新技术，试图提高演绎能力。第一个是在求解过程中执行的检查。二是将简化与提高演绎能力相结合的基于精馏的预处理算法。 目前大多数SAT解算器在不同阶段采用分解来派生新的条款或简化现有条款。前者发生在冲突分析过程中，后者通常在预处理过程中进行。我们展示了如何在解析过程中低成本地检测解析器对操作数的包含;然后，我们讨论了如何使用这种检测来改进SAT求解器的三个阶段:变量消除、子句蒸馏和冲突分析。“即时”的包容检查很容易集成在SAT求解器中。特别地，它兼容强冲突分析和不满意证明的生成。实验证明了新技术的有效性。索引术语:cnf，蒸馏，DPLL，预处理，SAT。
	I. Introduction The last two decades have seen great advances in the performance of satisfiability (SAT) solvers for propositional logic, in particular those based on the David–Putnam– Logemann–Loveland (DPLL) procedure [1]–[5]. These solvers have evolved in symbiotic relationship with many EDA applications including model checking, logic synthesis, testing, and timing analysis. Progress has been made both in the pruning of the search space [3] and in the efficient implementation of the basic operations, like deductions [4]. Here we are concerned with techniques that transform a conjunctive normal form (CNF) formula, either as a preprocessing step [6]–[8] or during the DPLL procedure. These transformations should be relatively inexpensive and produce formulae on which the DPLL procedure runs faster than on the original ones. 在这里，我们关注的是转换合取范式(CNF)公式的技术，无论是作为预处理步骤[6]-[8]还是在DPLL过程中。这些转换应该相对便宜，并且产生的公式使DPLL程序比原始程序运行得更快。 Reducing the size of the formula is a common objective of transformations. For instance, a set of clauses is redundant if a proper subset represents the same function. A subsumed clause (i.e., a clause implied by another) is redundant, and the cost of many SAT solver operations decreases with a smaller formula. Hence, removing subsumed clauses is usually beneficial. However, not all redundant clauses can be removed without negative effect on the speed of the solver. 例如，如果一组子句的适当子集表示相同的函数，则该子句集是冗余的。一个包含的子句(即，由另一个隐含的子句)是多余的，许多SAT求解器操作的成本减少了一个较小的公式。因此，去掉包含子句通常是有益的。然而，并不是所有的冗余子句都可以在不影响求解速度的情况下被删除。 We introduce two notions that help in the design and evaluation of formula transformations. The first is deductive power of a CNF formula. The higher this power, the more consequences the DPLL procedure can deduce from each of its decisions; hence, the more effective is the pruning of the search space. The second notion is proof conciseness. It reflects the fact that the DPLL procedure progresses through the search space by proving that parts of that space contain no satisfying assignment and recording such findings in the form of new clauses and their derivations. More concise proofs are faster to build and usually more effective at pruning further search. 我们介绍了两个概念，有助于设计和评估公式转换。第一个是CNF公式的演绎能力。该功率越高，DPLL程序可以从其每个决策中推断出的结果越多;因此，更有效的是对搜索空间进行修剪。第二个概念是证明简洁性。它反映了这样一个事实，即DPLL程序通过证明该空间的某些部分不包含令人满意的赋值，并以新条款及其衍生的形式记录这些发现，从而在搜索空间中进行进展。更简洁的证明可以更快地建立，并且通常更有效地减少进一步的搜索。 To see how deductive power may help in the analysis of SAT solvers, consider clause recording, which adds conflictlearned clauses or, simply, conflict clauses to the original SAT instance. Each conflicting assignment is analyzed to identify a subset that is sufficient to cause the current conflict. The disjunction of the literals in the subset becomes a new clause added to the original SAT instance. The conflict clauses learned by SAT solvers are by definition redundant, but they always improve the deductive power of a CNF formula. 要了解演绎能力如何有助于分析SAT解题，请考虑条款记录，它将冲突学习条款或简单地说，冲突条款添加到原始SAT实例中。对每个冲突分配进行分析，以确定足以引起当前冲突的子集。子集中字面值的分离成为添加到原始SAT实例中的新子句。SAT求解器学习的冲突子句从定义上讲是冗余的，但它们总是提高CNF公式的演绎能力。 Clauses that are subsumed by other clauses slow down the implication process, but do not help the solver in pruning the search space. We show that they never improve deductive power. Therefore, preprocessing often removes them to accelerate implications. On the other hand, removing literals from clauses may increase the deductive power of a formula. We study in detail several approaches to such elimination, both as preprocessing and during DPLL. 被其他子句所包含的子句减慢了隐含过程，但无助于求解器修剪搜索空间。我们表明，被包含的子句从来没有提高演绎能力。因此，预处理通常会删除它们以加速蕴含推导。另一方面，从子句中去掉字面可能会增加公式的演绎能力。我们详细研究了几种消除方法，包括预处理和DPLL过程。 Literal removal procedures are often based on resolution. In addition, resolution may be applied to eliminate variables from the formula. Since the elimination of variables may increase the number of clauses, it is usually applied with restraint [6], [9]. Deductive power is not guaranteed to improve either. Instead, the main benefit of variable elimination is the decrease in the average number of decisions and implications required to produce a conflicting assignment. Not only conflicts occur sooner, but their analysis is faster, and the learned clauses tend to prune larger portions of the search space. 此外，还可以应用解析来消除公式中的变量。由于消除变量可能会增加子句的数量，因此通常是有限制地使用它。   In this paper, we analyze existing techniques that increase deductive power or generate more concise implication graphs and we propose two new ones. We show how to detect subsumptions during resolution during both preprocessing and conflict analysis with minimal overhead. Our on-the-fly subsumption check can be applied to both regular and strong [10] conflict analysis. We show how this inexpensive check is used to improve deductive power at three stages of the SAT solver: variable elimination, clause distillation, and conflict analysis. We then describe a distillation algorithm that asserts the negations of clauses to remove redundant literals and possibly derive new clauses. Unlike previous approaches, this distillation procedure may replace a clause with the resolvent of two or more existing clauses without explicitly deriving any such resolvents in advance. We show that distillation increases deductive power and shortens implication graphs. 我们分析了现有的提高演绎能力或生成更简洁蕴涵图的技术，并提出了两种新的技术。我们将展示如何在预处理和冲突分析的解决过程中以最小的开销检测包含。我们的动态包容检查既可以应用于常规冲突分析，也可以应用于强冲突分析[10]。我们展示了如何在SAT求解的三个阶段使用这种廉价的检查来提高演绎能力:变量消除、条款蒸馏和冲突分析。然后，我们描述了一种蒸馏算法，该算法断言子句的否定以删除冗余的文字并可能派生出新的子句。与以前的方法不同，这种蒸馏程序可以用两个或两个以上现有子句的解来代替一个子句，而无需事先明确地推导出任何这样的解。我们证明蒸馏提高了演绎能力，缩短了蕴涵图。 Experiments show that the presented techniques speed up our SAT solver. Variable elimination works primarily by shortening the implication graphs, while other transformations mainly improve deductive power. 实验表明，所提出的方法提高了求解速度。变量消除主要通过缩短蕴涵图来实现，而其他变换主要是提高演绎能力。 This paper combines and extends [11] and [12]. It is organized as follows. Section II discusses related work. Section III covers background. In Section IV we introduce and characterize the notion of deductive power of a CNF formula. In Section V we describe our on-the-fly simplification based on self-subsumption during conflict analysis and present the details of the algorithm. Section VI describes our distillationbased approach. Section VII reports results from a prototype implementation. We draw conclusions and outline future work in Section VIII. 本文对[11]和[12]进行了合并和推广。第四章节我们引入并描述了CNF公式的演绎强度的概念；第五章节在冲突分析过程中基于自包容的动态简化，并给出了算法的细节。第六章节描述了我们基于蒸馏的方法。第七章节报告了原型实现的结果。
	II. Related Work A problem related to preprocessing of a CNF formula is the preprocessing of conflict clauses in an incremental SAT solver. An incremental solver is given a sequence of SAT instances and tries to use clauses learned in earlier instances to expedite the solution of later instances. If each instance is obtained from the previous by addition of new clauses, all clauses learned by the solver can be forwarded to the new instance. However, in the general case, clauses must be validated before they can be forwarded. In [13], a process called distillation was proposed, which forwards a clause derived from a previously learned clause γ only if asserting the negation of γ causes a conflict in the new instance. In [11] and in this paper, we apply distillation to preprocessing the original clauses of a CNF formula and we characterize this approach from the point of view of deductive power. 增量求解器给出一系列SAT实例，并尝试使用在早期实例中学习到的子句来加速后续实例的求解。如果每个实例都是通过添加新的子句来获得的，那么求解器学习到的所有子句都可以转发到新的实例中。在[13]中，提出了一种称为蒸馏的过程，仅当断言γ的否定导致新实例中的冲突时，该过程才会转发从先前学习的子句γ派生的子句。在[11]和本文中，我们将蒸馏应用于预处理CNF公式的原始分句，并从演绎能力的角度描述了这种方法。 Assignment shrinking [14] can also be seen as on-the-fly distillation of selected conflict clauses. At the end of conflict analysis, the algorithm of [14] backtracks to a level preceding the backtracking level to undo some assignments in the conflict clause. It then applies those assignments again in a different order until a new conflict occurs. This may produce a new smaller conflict clause. Since this is a potentially expensive technique, its invocation is controlled by a heuristic. 分配缩减[14]也可以看作是对选定的冲突条款的动态提炼。在冲突分析结束时，[14]算法回溯到回溯层之前的一层，撤销冲突子句中的一些赋值。然后，它以不同的顺序再次应用这些分配，直到出现新的冲突。这可能会产生一个新的较小的冲突条款。由于这是一种潜在的昂贵技术，因此它的调用由启发式控制。 Previous work besides [14] has addressed the quality of conflict clauses [5], [7], [10], [15], [16]. In particular, the clause minimization algorithm of [7] and [16] traverses the implication graph beyond the 1-unique implication point (UIP) to remove literals in the conflict clause that are implied by other literals. The strong conflict analysis proposed in [10] generates a second conflict clause that is often more effective than a regular conflict clause of [15] in escaping regions of the search space where the solver would otherwise linger for a long time. A common thread of most work on the subject is the search for a balance between a technique’s cost and its ability to detect implications earlier. Unlike the on-the-fly subsumption to be discussed in Section V, these earlier techniques focus on simplification of the conflictlearned clauses, instead of looking at all clauses appearing in the resolution graph. 除[14]外，先前的工作已经解决了冲突条款的质量问题。特别是，[7]和[16]的子句最小化算法在1唯一隐含点(UIP)之外遍历隐含图，以删除冲突子句中由其他字面量隐含的字面量。[10]中提出的强冲突分析生成的第二个冲突子句通常比[15]中的常规冲突子句更有效，因为在搜索空间的转义区域，求解器将在该区域停留很长时间。关于这个主题的大多数工作的一个共同主线是寻找一种技术的成本和早期发现影响的能力之间的平衡。与第五节中讨论的即时包含不同，这些早期的技术侧重于简化冲突学习子句，而不是查看解决图中出现的所有子句。 An existing clause may be subsumed by a conflict clause newly found by any of the conflict analysis algorithms. Hence, one may try to simplify the newly redundant clauses. The on-the-fly simplification algorithm used in [17] can detect the subsumed clause with a one watched literal scheme, when a new clause is generated by conflict analysis. While the one watched literal scheme is efficient, the removal of subsumed clauses does not improve deductive power and does not produce more concise proofs. The practical ability of this technique to speed up SAT solvers was not the focus of [17] and remains to be established. 现有子句可以被任何冲突分析算法新发现的冲突子句所包含。因此，人们可以设法简化新出现的冗余从句。[17]中使用的实时简化算法可以在冲突分析生成新子句时，以一个监视的文字方案检测包含子句。虽然一个人看到的字面方案是有效的，但删除包含的条款并不能提高演绎能力，也不能产生更简洁的证明。该技术加速SAT求解的实际能力不是[17]的重点，仍有待建立。
	III. Preliminaries
	IV. Deductive Power of a CNF Formula                           The deductive power of Definition 1 is related to, but distinct from the deducibility of [20], which counts the number of implications due to assignment to a variable of a CNF formula. 定义1的演绎能力与之相关，但与之不同可演绎性[20]，其中计算了由于的含义数量对CNF公式的变量赋值。   While adding a transitive closure clause of the implications does not affect deductive power, it may help the solver by shortening the implication graph. A more concise implica tion graph may benefit the procedures that work on it. For instance, the deduction procedure may identify a conflicting clause more quickly, and conflict analysis may resolve fewer antecedents. On the other hand, adding clauses to the CNF database indiscriminately may substantially slow down the deduction procedure. To prevent this, a supplemental clause should be generated only when its usefulness is established by an effective criterion (i.e., strong conflict analysis).  虽然添加隐含的传递闭包子句不会影响演绎能力，但它可以通过缩短隐含图来帮助求解器。更简洁的隐含图可能有利于处理它的程序。例如，演绎程序可以更快地识别冲突条款，而冲突分析可以解决更少的前因。 另一方面，不分青红皂白地向CNF数据库添加条款可能会大大减慢扣除程序。为了防止这种情况，只有当其有用性由有效标准（即强有力的冲突分析）确定时，才应生成补充条款。
	V. On-The-Fly Self-Subsumption Lemma 1 implies that simplification based on self subsumption may improve the deductive power of a CNF formula. Since detecting whether the resolvent of two clauses subsumes either operand is easy and inexpensive, checking on the-fly for subsumption can be added with almost no penalty to those operations of SAT solvers that are based on resolution. In this section we review the basic idea and detail the application of the on-the-fly subsumption check to conflict analysis. Then, we discuss on-the-fly subsumption in preprocessing. An efficient on-the-fly check for subsumption during reso lution is based on the following elementary fact.
	VII. Experimental Results     We have presented techniques that aim at increasing the deductive power of a CNF formula and promoting more concise implication graphs. In order to evaluate them, we have implemented a preprocessor on top of the CNF SAT solver CirCUs 2.0 [23], [24], which applies variable elimination, the distillation procedure of Section VI, named Alembic, and simplification based on subsumption and self-subsumption as in [6]. We have also implemented the three applications of onthe-fly clause simplification discussed in this paper, namely, to variable elimination and conflict analysis in Alembic as well as to conflict analysis in CirCUs. In variable elimination, an increase in the average length of the clauses is detrimental for deductive power. Hence, in our implementation, only variables whose elimination does not cause such an increase are eliminated. Since SAT solvers often need to provide either a satisfying assignment or a proof of unsatisfiability, clauses that are either removed or simplified are set aside just as the derivations of conflict clauses [25], [26]. The SAT solver CirCUs only needs these clauses to recover a complete solution (for a satisfiable instance), or to produce a proof of unsatisfiability in terms of the original clauses. This scheme requires extra memory, but its effect on speed is negligible. The benchmark suite is composed of all the CNF instances (with no duplicates) from the industrial category of the SAT Races of 2006 and 2008, and the SAT Competitions of 2007 and 2009 [27]. We conducted the experiments on a 2.4 GHz Intel Core2 Quad processor with 4 GB of memory. We used 10,000 s as timeout, and 2 GB as memory bound. We tested MiniSat 2.0 [19] and PrecoSAT 236 [28] along with CirCUs 2.0 to provide reference points. The plot of Fig. 19 shows how many instances are solved by selected solvers within a given time bound. Our variable elimination algorithm is named EV; Alembic is abbreviated AL, EVAL stands for EV + AL, and OCI denotes the onthe-fly clause improvement described in Section V. Fig. 19 shows the CPU time taken by CirCUs (with various subsets of the proposed approaches), MiniSat, and PrecoSAT. Both MiniSat and PrecoSAT use their own preprocessors [6]. Fig. 19 confirms that CirCUs is comparable to state-of-the-art SAT solvers, and that its performance is significantly improved by applying all the proposed approaches (i.e., EVAL + OCI). The scatterplots of Fig. 20 examine the effects of the proposed techniques on deductive power and size of implication graphs, by showing the changes in CPU time, numbers of decisions, average numbers of resolution steps per conflict analysis, and average length of conflict-learned clauses. For each of these quantities the geometric mean of the new/old ratios is reported (excluding cases in which one of the values is 0). Single-sample t-tests were performed to confirm the statistical significance of the data. The null hypothesis was that the mean of the logarithms of the ratios is 0. The alternative hypothesis is two-sided. Since the data that are compared span several orders of magnitudes, differences and ratios may paint very different pictures of the experiments. Analyzing the ratios puts equal emphasis on short and long-running instances. This is partly compensated by the scatterplots and the views in Fig. 19, which highlight the ability of the improved procedure to complete more instances in the allotted time. A marked decrease in the numbers of decisions confirms that the proposed techniques allow the SAT solver to rely more on deduction and less on search. The reduction in resolution steps confirms that the implication graphs are, on average, significantly smaller. As a result, shorter clauses are learned. For lack of space, we omit scatterplots illustrating the effects of individual techniques. They would show that variable elimination is the main cause for the smaller implication graphs, and that it also tends to reduce the number of decisions and shorten the learned clauses. Distillation alone decreases the numbers of decisions (as one would expect of a technique addressing deductive power) and shortens learned clauses, but has limited effect on the sizes of the implication graphs. Its effect on memory consumption proves negligible. Variable elimination interacts in an interesting way with OCI. This is shown in Fig. 21, where the numbers of on-thefly subsumptions per resolution step during DPLL are seen to increase significantly when EV is applied. The following example sheds light on this phenomenon. Example 14: Consider the following clauses:                           Fig. 23 shows that the conflict clause subsumes c2. (It also subsumes c6, but this is not detected by the algorithm.) This time there are fewer resolution steps, and this “abridgment” of the process allows the subsumed clause to enter the analysis right before the subsuming resolvent is computed instead of several steps before. We now report statistics on the performance of the preprocessors. Fig. 24 compares the speed of various versions of EVAL to SatELite. (In these plots, SatELite is run on all CNF formulae, while, in Fig. 19, the solver may disable SatELite depending on the size of CNF formula.) OCI contributes to the improved preprocessor speed. This is clear in the case of EVAL versus EVAL + OCI. It is true also without distillation, because EV + OCI removes significantly more clauses and literals than plain EV in about the same time. It is also interesting to compare the reductions achieved by different preprocessors. In Fig. 25, we report the fractions of instances that achieve certain reductions in terms of variables, clauses, and literals. About 10% of the instances achieve close to 100% reduction. This means that preprocessing reduces the CNF formulae to either the empty clause or the empty set of clauses. CirCUs’s variable elimination is less aggressive than SatELite’s: it eliminates fewer clauses, but almost never increases the number of literals. Adding Alembic yields the least number of clauses without compromising the good performance in terms of literals. While conflict analysis during distillation may produce additional conflict clauses, the number of added clauses is on average 0.1% of the total. Alembic often achieves more simplifications thanks to the on-the-fly subsumption check. The mean number of clauses simplified per conflict is 0.7. Moreover, on average, in 51% of the conflicts the 1-UIP clauses subsumes one of the clauses used to resolve it; in those cases, rather than the 1-UIP clause being added to the database, the operand is simplified.
	VIII. Conclusion We have presented efficient transformations of a CNF formula that aim at either improving its deductive power or shortening implication graphs. We have shown that the transformations help a DPLL-based SAT solver to run faster by deducing more literals from its decisions and by reducing the depth of the implication graphs used in conflict analysis. On-the-fly simplification based on self-subsumption can be applied to any stage that uses resolution, e.g., conflict analysis and variable elimination, with minimal overhead. Its application is compatible with advanced conflict analysis techniques and with the generation of unsatisfiability proofs. Another benefit is the reduction of the number of added conflict clauses without detriment for the deductive power. The distillation procedure applied to preprocessing of the CNF formula also considerably speeds up the SAT solver by increasing deductive power. In contrast, we have shown that variable elimination works mainly by reducing the number of resolution steps required in conflict analysis. This results in earlier conflicts, cheaper analyses and better conflict clauses. The proposed techniques have several other extensions that we plan to investigate: generation of small unsatisfiable cores, application to restarts and solution enumeration, application to non-clausal reasoning, and logic synthesis and representation of sets by characteristic functions in CNF [29].
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