蕴含图-冲突分析-强冲突分析

蕴含图文献：按时间顺序

1.  URL:http://vlsi.colorado.edu/~vis.

2. L. Zhang and S. Malik. Validating SAT solvers using an independent resolution-based checker: Practical implementations and other applications. In Design, Automation and Test in Europe (DATE03), pages 880-885, Munich, Germany, Mar. 2003.

 

4. HoonSang Jin and F. Somenzi, "Strong Conflict Analysis for Propositional Satisfiability," Proceedings of the Design Automation & Test in Europe Conference, Munich, Germany, 2006, pp. 1-6, doi: 10.1109/DATE.2006.244149.

5. HyoJung Han, Fabio Somenzi, HoonSang Jin:
Making Deduction More Effective in SAT Solvers. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 29(8): 1271-1284 (2010)

 

	SAT求解器Grasp [1]使用冲突子句(称为无商品)来引导搜索空间的探索。包含唯一蕴涵点(UIP)的子句在回溯之后变得断言，因此保证了对可能的解决方案的检查的进展。   [2]的作者研究了从导致冲突的蕴涵链(所谓的冲突蕴涵图)中提取包含UIP的子句的几种方法，并发现选择第一个UIP(即最接近冲突的那个)效果最好。

 

 

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文献1介绍了基本概念、蕴含图、冲突分析并提到了强冲突分析，文献2重点讲解了强突分析。

1.文献1：Making Deduction More Effective in SAT Solvers

	When the 1-UIP is far from the conflict in the implication graph, the conflict clause may not be effective in preventing the SAT solver from producing many conflicts involving the same clause. 当 1-UIP 与隐含图中的冲突相距甚远时，冲突子句可能无法有效阻止 SAT 求解器产生涉及同一子句的许多冲突。   Strong conflict analysis [10] can be a remedy in such cases: It examines intermediate resolvents as UIP based conflict analysis does. Contrary to UIP-based analysis, however, it generates an additional conflict clause that contains more than one literal assigned at the current decision level. This additional conflict clause must be one of the intermediate resolvents derived between the conflict and the 1-UIP. Usually, the closer to the conflict, the fewer literals the resolvent contains. Therefore, the additional conflict clause tends to be shorter than the conflict clause with the 1-UIP. 在这种情况下，强冲突分析[10]可以作为一种补救措施：它像基于UIP的冲突分析一样检查中间解决方案。然而，与基于 UIP 的分析相反，它会生成一个额外的冲突子句，其中包含在当前决策级别分配的多个文本。 此附加冲突条款必须是冲突和 1-UIP 之间派生的中间解决方案之一。通常，越接近冲突，解决方案包含的文字就越少。因此，附加冲突子句往往比具有 1-UIP 的冲突子句短。

 

文献2：Strong conflict analysis for propositional satisfiability

• HoonSang Jin, Fabio Somenzi:
  Strong conflict analysis for propositional satisfiability. DATE 2006: 818-823

	In difficult SAT instances, solvers often spend inordinate amounts of time on comparatively small fractions of the search space. We call such small sets of configurations hot spots. The existence and location of such hot spots obviously depends on the SAT algorithm and one objective of an effective conflict analysis procedure is to reduce their occurrence as much as possible by adding appropriate conflict clauses. Though clauses based on first UIPs are better than other UIP-based clauses, they do not directly address the issue of hot spots. 在困难的SAT案例中，解答者经常在搜索空间的相对小的部分上花费过多的时间。我们称这种小的组态集为热点。这种热点的存在和位置显然取决于SAT算法，并且有效的冲突分析过程的一个目标是通过添加适当的冲突子句来尽可能地减少它们的发生。虽然基于第一次UIP的条款比其他基于UIP的条款更好，但它们没有直接解决热点问题。
	正如不同的子句可以从相同的蕴涵图中导出一样，不同的蕴涵图可以从相同的决策序列中获得，这取决于蕴涵传播的顺序。最近的两篇论文提出通过修改SAT求解器产生的蕴涵图来提高冲突子句的质量。[4]的作者指出，即使冲突分析可能从相同的冲突子句开始，它也可能在不同的蕴涵图上产生不同的冲突学习子句。他们提出了一种方法，如果发现了更小的子句，则更新变量的前件，同时传播含义，以便冲突分析可以找到更小的冲突子句。
	正如不同的子句可以从相同的蕴涵图中导出一样，不同的蕴涵图可以从相同的决策序列中获得，这取决于蕴涵传播的顺序。   最近的两篇论文提出通过修改SAT求解器产生的蕴涵图来提高冲突子句的质量。 [4]的作者指出，即使冲突分析可能从相同的冲突子句开始，它也可能在不同的蕴涵图上产生不同的冲突学习子句。他们提出了一种方法，如果发现了更小的子句，则更新变量的前件，同时传播含义，以便冲突分析可以找到更小的冲突子句。     [4] Z. Fu, Y. Mahajan, and S. Malik. New features of the S AT04 versions of zChaff. SAT Competition 2004 - Solver Description, May 2004
	为此，它回溯足够长的时间来撤消对子句中文字的所有赋值，然后只重新应用子句中的赋值。这为相同的冲突创建了一个新的、通常更小的蕴涵图。不幸的是，收缩是一个非常昂贵的操作，因为需要大量的回溯，而且它不能保证文字数量的减少。多重冲突分析是增强冲突分析的另一种方式。然而，这是一种昂贵的技术，应该有一个良好的标准来指导，尽可能限制其应用于无利可图的情况。
	In our experience, techniques that restructure the implication graph must be used sparingly because they tend to be expensive and they are often ineffective. 根据我们的经验，重构蕴涵图的技术必须谨慎使用，因为它们往往是昂贵的，而且通常是无效的。
	In this paper we focus instead on a stronger conflict analysis, one that often learns better clauses than those including the first UIP. We observe that there are many cases when multiple conflicts occur on one clause. A clause generated by conflict analysis prevents the same conflict from occurring again. However, the current conflict analysis is not strong enough to prevent further conflicts on the same clause. When this happens, the clauses learned through conflict analysis are often ineffective at pruning the search. 在本文中，我们将重点放在更强的冲突分析上，这种分析通常比包含第一个UIP的分析学习到更好的从句。我们注意到，在很多情况下，一个子句会发生多个冲突。冲突分析生成的子句可防止相同的冲突再次发生。然而，当前的冲突分析还不足以防止同一条款上的进一步冲突。当这种情况发生时，通过冲突分析学习到的子句在修剪搜索时通常是无效的。
	There is a reasonable number of cases  in which many conflicts happen on the same clause. Our approach  learns two clauses for such conflicts. One clause contains the first  UIP : Its main responsibility is to drive the search forward. The second clause is devoted to a better pruning of yet-to-be-visited parts  of the search space.  It is generated when the first clause is likely  to be ineffective and the chance of repeated conflicts on the same  clause is deemed high. 在合理数量的案例中，同一条款上发生了许多冲突。我们的方法学习了针对此类冲突的两个子句。一个子句包含第一个 UIP：它的主要职责是推动搜索向前发展。第二个子句致力于更好地修剪搜索空间中尚未访问的部分。 当第一个子句可能无效并且被认为在同一子句上重复冲突的可能性很高时，就会产生它。
	We have presented a strong conflict analysis procedure for propositional satisfiability that adds more than one conflict clause from one conflict. We have shown that the additional conflict clauses help reduce the number of literals in conflict clauses as well as the depth of the implication graphs. The experimental results show large improvements compared to the state-of-the-art SAT solvers. Experimental evidence supports the claim that the performance gains are indeed due to the improved conflict analysis. 我们已经为命题可满足性提出了一个强冲突分析过程，它从一个冲突中增加了一个以上的冲突子句。我们已经表明，额外的冲突子句有助于减少冲突子句中的文字数量以及蕴涵图的深度。实验结果显示，与最先进的SAT求解器相比，有了很大的改进。实验证据支持这样的说法，即性能的提高确实是由于改进了冲突分析。     Several details of the procedure warrant further study. For instance, even though we have devised a criterion to apply the shrinking method based on the occurrence of conflicts on the same clause, the technique remains expensive in term of CPU time, and we need to investigate a more efficient way of applying it. 该程序的几个细节值得进一步研究。例如，即使我们已经设计了一个基于同一子句上冲突的发生来应用收缩方法的标准，该技术在CPU时间方面仍然是昂贵的，并且我们需要研究一种更有效的方法来应用它。     Random restart techniques have been used to enhance several SAT solvers. We conjecture that deep implication graphs and multiple conflicts on same clause make the search dwell in hot spots. Therefore in some case, restarts help the search out of a hot spot. Our strong conflict analysis should reduce the need for such random restarts. 随机重启技术已经被用于增强几个SAT求解器。我们推测深层蕴涵图和同一子句上的多重冲突使得搜索停留在热点上。因此在某些情况下，重启有助于搜索出热点。我们强大的冲突分析应该减少这种随机重启的需要。