文献学习-On CDCL-based proof systems with the ordered decision strategy

On CDCL-based proof systems with the ordered decision strategy

基于cdcl的基于有序决策策略的证明系统

Nathan Mull∗   Shuo Pang†    Alexander Razborov‡
September 11, 2019 

∗University of Chicago, Department of Computer Science, nmull@cs.uchicago.edu.
†University of Chicago, Department of Mathematics, spang@math.uchicago.edu.
‡University of Chicago, USA, razborov@math.uchicago.edu and Steklov Mathematical Institute, Moscow, Russia, razborov@mi-ras.ru.

Cite this paper as:

Mull N., Pang S., Razborov A. (2020) On CDCL-Based Proof Systems with the Ordered Decision Strategy. In: Pulina L., Seidl M. (eds) Theory and Applications of Satisfiability Testing – SAT 2020. SAT 2020. Lecture Notes in Computer Science, vol 12178. Springer, Cham. https://doi-org-s.era.lib.swjtu.edu.cn/10.1007/978-3-030-51825-7_12

 

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Abstract

	We prove that conflict-driven clause learning SAT-solvers with the ordered decision strategy and the DECISION learning scheme are equivalent to ordered resolution. We also prove that, by replacing this learning scheme with its opposite that stops after the first new clause when backtracking, it becomes equivalent to general resolution. To the best of our knowledge, this is the first theoretical study of the interplay between specific decision strategies and clause learning. 译文：我们证明了冲突驱动子句学习sat求解器与有序决策策略和决策学习方案是等价的. 译文：我们还证明，用回溯时在第一个新子句后停止的学习方案替换这个学习方案，它等价于一般的解析。 译文：据我们所知，这是第一个关于具体决策策略和子句学习之间相互作用的理论研究。
	For both results, we allow nondeterminism in the solver’s ability to perform unit propagation, conflict analysis, and restarts, in a way that is similar to previous works in the literature. 译文：对于这两个结果，我们允许求解器在执行单元传播、冲突分析和重新启动的能力中存在不确定性，其方式类似于文献中以前的工作。To aid the presentation of our results, and possibly future research, we define a model and language for discussing CDCL-based proof systems that allows for succinct and precise theorem statements.译文：为了帮助展示我们的结果，以及可能的未来研究，我们定义了一个模型和语言来讨论基于cdcl的证明系统，它允许简洁和精确的定理陈述。

1. Introduction

SAT-solvers have become standard tools in many application domains such as hardware verification, software verification, automated theorem proving, scheduling and computational biology (see [24, 26, 16, 31, 19] among the others). Since their conception in the early 1960s, More recently, Vinyals [20] has shown that CDCL with the VSIDS decision strategy – among other common dynamic decision strategies – does not simulate general resolution.  译文：最近，Vinyals[20]已经表明，在其他常见的动态决策策略中，使用VSIDS决策策略的CDCL不能模拟一般的归结。  We attempt to make progress on this question by studying a simple decision strategy that we call the ordered decision strategy.  译文：我们试图通过研究一种简单的决策策略来解决这个问题，我们称之为有序决策策略。   This strategy is identical to the one studied by Beame et al. [4] in the context of DPLL without clause learning.译文：该策略与Beame等人在不带子句学习的DPLL环境下研究的策略相同。 It is defined naturally: when the solver has to choose a variable to assign, it chooses the smallest unassigned variable according to some fixed order and chooses its assigned value nondeterministically.译文：它的定义很自然:译文：当求解器需要选择一个变量进行赋值时，它按照一定的顺序选择最小的未赋值变量，并非确定性地选择赋值变量。 If unit propagation is used, the solver may assign variables out of order; a unit clause does not necessarily correspond to the smallest unassigned variable.译文：如果使用单位传播，求解器可能会无序分配变量;unit子句并不一定对应于最小的未分配变量。 This possibility of “cutting the line” is precisely what makes the situation more subtle and nontrivial. Thus, our motivating question is the following:译文：这种“断线”的可能性正是使情况更加微妙和重要的原因。因此，我们的激励问题如下:

 

5 Conclusion

	Our work continues the line of research aimed at better understanding theoretical limitations of CDCL solvers. We have focused on the impact of decision strategies, and we have considered the simple strategy that always chooses the first available variable under a fixed ordering. We have shown that, somewhat surprisingly, the power of this model heavily depends on the learning scheme employed and may vary from ordered resolution to general resolution. 译文：我们的工作继续着旨在更好地理解CDCL解决器的理论局限性的研究。我们已经关注了决策策略的影响，并且我们已经考虑了一个简单的策略，在一个固定的顺序下总是选择第一个可用的变量。我们已经证明了，有点令人惊讶的是，这个模型的能力很大程度上依赖于所采用的学习方案，并且可能在有序归结和一般归结之间有所不同。
	In practice, the fact that CDCL(DECISION-L,π-D,ALWAYS-C,ALWAYS-U)CDCL(DECISION-L,π-D,ALWAYS-C,ALWAYS-U) is not as powerful as resolution supports the observation that CDCL solvers with the ordered decision strategy are often less efficient than those with more powerful decision strategies like VSIDS. But, although DECISION-LDECISION-L is an asserting learning strategy, most solvers use more efficient asserting strategies like 1-UIP. A natural open question then is what can be proved if DECISION-LDECISION-L is replaced with some other amendment modeling a different, possibly more practical asserting learning scheme? Furthermore, what is the exact strength of CDCL(π-D,ALWAYS-C,ALWAYS-U)CDCL(π-D,ALWAYS-C,ALWAYS-U)? 译文：在实践中，CDCL(decision - l, rd -D,ALWAYS-C,ALWAYS-U)不如resolution强大，这一事实支持这样的观察，即使用有序决策策略的CDCL求解器的效率往往低于使用更强大决策策略(如VSIDS)的求解器。
	需要考虑到在搜索空间中搜索是如何发展到当前冲突点的——过去的信息需要进一步挖掘和整理。 另一个问题的时间与空间复杂度。最新的策略都不涉及全体变量和全体子句，值关注对当前冲突生成产生影响的相关信息。被抛弃的策略如-- DLIS分数计算为出现字面值时仍未满足的子句数。A typical example is the dynamic literal individual sum heuristic (DLIS). It selects as next decision literal one with the largest DLIS score, which is computed as the number of still unsatisfied clauses in which a literal occurs. A well-known and often applied variant of DLIS is the Jeroslow- Wang heuristic [20], which for instance is discussed in [21], together with other related early decision heuristics, including Bohm’s, MOM’s, etc.