基于聚类和划分的SAT分治判定

提出了一种将布尔公式划分为子句组来进行布尔可满足性判定的方法.CNF(conjunctive normal form)公式是可满足的当且仅当划分产生的每个子句组都是可满足的,因此,通过判定子句组的可满足性来判定原公式的可满足性,相当于用分治法将复杂问题分解为多个子问题来求解.这种分治判定方法一方面降低了原公式的可满足性判定复杂度;另一方面,由于子句组的判定可以并行,因而判定速度能够得到进一步的提高.对于不能直接产生布尔子句组划分的情形,提出了一种利用聚类技术将CNF公式聚类成多个簇,然后消去簇间的公共变量来产生子句组划分的方法.     

一阶子句搜索方法

子句集的可满足性判定是自动证明领域的热点之一.提出了子句搜索方法判定命题子句集Φ的可满足性,该方法查找Φ中子句的一个公共不可扩展子句C,当且仅当找到C时Φ可满足,此时C中各文字的补构成一个模型.结合部分实例化方法将子句搜索方法提升至一阶.一阶子句搜索方法可以判定子句集的M可满足性,具备终止性、正确性和完备性,是一种判定子句集可满足性的有效方法.     

一种基于伪布尔可满足性FPGA布线算法

为了克服布尔可满足性算法在现场可编程门阵列布线中存在的不足,引进了一种在标准对称阵列(隔离岛状)现场可编程门阵列结构下的新型有效布线方法——伪布尔可满足性算法,并结合实例详细地阐述了将其应用于布线的原理及方法,同时采用实际工业电路将布尔可满足性算法与伪布尔可满足性算法作出比较。实验结果显示,伪布尔可满足性算法比布尔可满足性算法在布线时间上减少了10.5%,在稳定性上提高了3.3%。     

一种布尔公式的代数逻辑约化新方法

布尔可满足问题是最早被证明的NP完全问题之一,1-in-3-SAT问题是一个NP完全的布尔可满足子类问题。1-in-3-SAT的计算复杂度取决于对应公式的变量以及子句的个数。将1-in-3公式归约为一个变量数或者子句数更少的1-in-3公式,是提高1-in-3-SAT问题求解效率的一个关键。基于一个新的范式形式——XCNF,针对1-in-3-SAT问题提出一种新的代数逻辑约化方法,用于在多项式时间内约减一个1-in-3公式的变量数和子句数。所提算法的主要思想为:首先将1-in-3公式转化为XCNF公式,然后尝试找出XCNF公式中的X-纯文字,并利用X-纯文字法则对1-in-3公式中相应的布尔变量赋值,最后得到一个约减公式,该约减公式与原公式的1-in-3可满足性等价。     

基于半扩展规则的定理证明方法

自动定理证明一直是人工智能领域中最重要的问题之一,基于归结的方法是通过推出空子句的方法来判定子句集的可满足性.基于扩展规则的定理证明方法在一定意义上是和归结原理对偶的方法,是通过子句集能否推导出所有极大项组成的子句集来判定可满足性.通过对扩展规则的研究给出了半扩展规则的概念,并提出了基于半扩展规则的定理证明算法SER.然后分析及证明了该算法的正确性、完备性和复杂性.实验结果表明,算法SER的执行效率较基于归结的有向归结算法DR和基于扩展规则算法IER,NER有明显的提高.     

基于子句文字长度动态约束的变量消除算法

变量消除算法作为一种重要的预处理算法已经应用于多种预处理器中。对比研究了在不同约束条件下,变量消除算法对简化性能和求解性能的影响,提出了基于子句文字长度动态约束的变量消除算法。该算法只允许当变量分解后的子句文字长度比原有子句文字少时,执行变量消除替换操作。在此基础上实现了一个基于Mini Sat开源代码的可满足性问题预处理器Mini Sat BFS。实验结果表明,与现有的基于子句数目约束的算法相比,新算法不仅降低了子句、变量和文字的数目,而且缩短了预处理过程和求解过程的时间消耗。更重要的是,改进后的算法在限定时间内可以求解更多的可满足性问题。     

半扩展规则下分解的定理证明方法

基于扩展规则的定理证明方法在一定意义上是与归结原理对偶的方法,通过子句集能否推导出所有极大项来判定可满足性.IER(improved extension rule)算法是不完备的算法,在判定子句集子空间不可满足时,并不能判定子句集的满足性,算法还需重新调用ER(extension rule)算法,降低了算法的求解效率.通过对子句集的极大项空间的研究,给出了子句集的极大项空间分解后子空间的求解方法.通过对扩展规则的研究,给出了极大项部分空间可满足性判定方法PSER(partial semi-extension rule).在IER算法判定子空间不可满足时,可以调用PSER算法判定子空间对应的补空间的可满足性,从而得到子句集的可满足性,避免了不能判定极大项子空间可满足性时需重新调用ER算法的缺点,使得IER算法更完备.在此基础上,还提出DPSER(degree partial semi-extension rule)定理证明方法.实验结果表明:所提出的DPSER和IPSER的执行效率较基于归结的有向归结算法DR、IER及NER算法有明显的提高.     

基于几何规划的布尔可满足问题求解方法

布尔可满足问题是计算机科学中诸多领域的重要问题,它的快速求解具有十分重要的意义.将具有实际物理背景的Solar算法中的拟物算法与几何规划相结合,提出并实现了一种布尔可满足性问题的连续求解方法.经实验验证,这种算法对布尔可满足性问题的求解具有一定的实用价值.     

基于细胞元模型拓扑约束求解

在模型制造领域,对于拓扑约束的求解是一个比较新的课题,以往的研究一直局限在拓扑优化方面。而且对其应用也仅限于模型的定义方面,在模型的声明与约束求解方面却没有得到应用。文章提出一种基于细胞元模型拓扑约束求解方法,通过该方法可以确定模型拓扑声明的关系,文章假设一个模型是由一个或多个细胞元组成的,并且能够用这些细胞元的组合来表示,对模型进行拓扑约束求解就是用来确定细胞模型中的每个细胞元是否是全约束的。要做到这点,文章将每个细胞元用一个布尔变量表示,把拓扑约束问题映射成为布尔可满足性问题。再对新的问题进行求解,从而解决了模型的拓扑约束求解问题。     

基于布尔可满足性的组合电路ATPG算法

布尔可满足性被深入研究并广泛应用于电子设计自动化等领域。该文提出了一种基于布尔可满足性的组合电路ATPG改进算法。在采用当前最新布尔可满足性求解程序加速策略的基础上,比如冲突驱动训练、冲突导向回跳和重启动技术等,引入电路结构信息来实现基于结构的分支决策。通过新增的电路结构信息层,布尔可满足性求解程序只需稍加修改,就能利用和及时更新此信息。最后给出的实验结果表明了算法的可行性和有效性。     

Stochastic Boolean Satisfiability

Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSAT, which can function for probabilistic domains as SAT does for deterministic domains. It shows the connection between SSAT and well-studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSAT instances. These algorithms are validated on random SSAT formulae generated under the fixed-clause model. In spite of the large complexity gap between SSAT (PSPACE) and SAT (NP), the paper suggests that much of what we have learned about SAT transfers to the probabilistic domain.     

Relaxations of the Satisfiability Problem Using Semidefinite Programming

We derive a semidefinite relaxation of the satisfiability (SAT) problem and discuss its strength. We give both the primal and dual formulation of the relaxation. The primal formulation is an eigenvalue optimization problem, while the dual formulation is a semidefinite feasibility problem. We show that using the relaxation, a proof of the unsatisfiability of the notorious pigeonhole and mutilated chessboard problems can be computed in polynomial time. As a byproduct we find a new `sandwich" theorem that is similar to the sandwich theorem for Lovász' -function. Furthermore, the semidefinite relaxation gives a certificate of (un)satisfiability for 2SAT problems in polynomial time. By adding an objective function to the dual formulation, a specific class of polynomially solvable 3SAT instances can be identified. We conclude with discussing how the relaxation can be used to solve more general SAT problems and with some empirical observations.     

最坏情况下Min-2SAT问题的上界

最坏情况下MaxSAT问题上界的研究已成为一个热门的研究领域.与MaxSAT问题相对的是MinSAT问题,在求解某些组合优化问题时,将其转化为MinSAT问题比转化为MaxSAT问题有着更快的速度,因此对MinSAT问题进行研究.针对Min-2SAT问题提出算法MinSATAlg,该算法首先利用化简算法Simplify对公式进行化简,然后通过分支树的方法对不同情况的子句进行分支.从子句数目的角度分析算法的时间复杂度并证明Min-2SAT问题可在O(1.134 3m)时间内求解,对于每个变量至多出现在3个2-子句中的情况,得到最坏情况下的上界为O(1.122 5n),其中n为变量的数目.     

Configuration landscape analysis and backbone guided local search.: Part I: Satisfiability and maximum satisfiability

Boolean satisfiability (SAT) and maximum satisfiability (Max-SAT) are difficult combinatorial problems that have many important real-world applications. In this paper, we first investigate the configuration landscapes of local minima reached by the WalkSAT local search algorithm, one of the most effective algorithms for SAT. A configuration landscape of a set of local minima is their distribution in terms of quality and structural differences relative to an optimal or a reference solution. Our experimental results show that local minima from WalkSAT form large clusters, and their configuration landscapes constitute big valleys, in that high quality local minima typically share large partial structures with optimal solutions. Inspired by this insight into WalkSAT and the previous research on phase transitions and backbones of combinatorial problems, we propose and develop a novel method that exploits the configuration landscapes of such local minima. The new method, termed as backbone-guided search, can be embedded in a local search algorithm, such as WalkSAT, to improve its performance. Our experimental results show that backbone-guided local search is effective on overconstrained random Max-SAT instances. Moreover, on large problem instances from a SAT library (SATLIB), the backbone guided WalkSAT algorithm finds satisfiable solutions more often than WalkSAT on SAT problem instances, and obtains better solutions than WalkSAT on Max-SAT problem instances, improving solution quality by 20% on average.     

由一阶逻辑公式得到命题逻辑可满足性问题实例

命题逻辑可满足性(SAT)问题是计算机科学中的一个重要问题.近年来许多学者在这方面进行了大量的研究,提出了不少有效的算法.但是,很多实际问题如果用一组一阶逻辑公式来描述,往往更为自然.当解释的论域是一个固定大小的有限集合时,一阶逻辑公式的可满足性问题可以等价地归约为SAT问题.为了利用现有的高效SAT工具,提出了一种从一阶逻辑公式生成SAT问题实例的算法,并描述了一个自动的转换工具,给出了相应的实验结果.还讨论了通过增加公式来消除同构从而减小搜索空间的一些方法.实验表明,这一算法是有效的,可以用来解决数学研究和实际应用中的许多问题.     

A Comparison between SAT and CSP Techniques

The aim of this paper is to establish a connection between the propositional logic and the constraint based reasoning frameworks. This work is based on a translation of the satisfiability problem (SAT) into the binary constraint-satisfaction problem (CSP). The structure of the SAT problem and its associated CSP are then exploited together for characterizing tractable SAT problems, increasing the effectiveness of the classical reduction rules: unit clause and monotone literal rules, and expressing the arc and path consistency concepts with logical inference rules. This study leads to compare the behaviors of the DP and MAC procedures for solving respectively a SAT instance and its binary CSP expression.     

EDA领域中可满足性问题求解方法研究

可满足性问题是理论计算机和人工智能中的著名问题,很多问题都可以通过可满足性求解方法解决.对EDA领域中可满足性问题的求解技术进行了研究.总结了目前主要的求解方法,并对不同的方法进行了详细的分类和比较.讨论了该领域研究中存在的问题,并指出了近期研究热点和未来发展趋势.     

On the Hybrid Propositional Encodings of Planning

Recently, casting planning as propositional satisfiability (SAT) has been shown to be an efficient technique of plan synthesis. This article is a response to the recently proposed challenge of developing novel propositional encodings that are based on a combination of different types of plan refinements and characterizing the tradeoffs. We refer to these encodings as hybrid encodings. An investigation of these encodings is important, because this can give insights into what kinds of planning problems can be solved faster with hybrid encodings.
Encodings based on partial–order planning and state–space planning have been reported in previous research. We propose a new type of encoding called a unifying encoding that subsumes these two encodings. We also report on several other hybrid encodings. Next, we show how the satisfiability framework can be extended to incremental planning. State–space encoding is attractive because of its lower size and causal encoding is attractive because of its highest flexibility in reordering steps. We show that hybrid encodings have a higher size and a lower flexibility in step reordering and, thus, do not combine the best of these encodings. We discuss in detail several specific planning scenarios where hybrid encodings are likely to be superior to nonhybrid encodings.     

Answer Set Programming Based on Propositional Satisfiability

Answer set programming (ASP) emerged in the late 1990s as a new logic programming paradigm that has been successfully applied in various application domains. Also motivated by the availability of efficient solvers for propositional satisfiability (SAT), various reductions from logic programs to SAT were introduced. All these reductions, however, are limited to a subclass of logic programs or introduce new variables or may produce exponentially bigger propositional formulas. In this paper, we present a SAT-based procedure, called ASPSAT, that (1) deals with any (nondisjunctive) logic program, (2) works on a propositional formula without additional variables (except for those possibly introduced by the clause form transformation), and (3) is guaranteed to work in polynomial space. From a theoretical perspective, we prove soundness and completeness of ASPSAT. From a practical perspective, we have (1) implemented ASPSAT in Cmodels, (2) extended the basic procedures in order to incorporate the most popular SAT reasoning strategies, and (3) conducted an extensive comparative analysis involving other state-of-the-art answer set solvers. The experimental analysis shows that our solver is competitive with the other solvers we considered and that the reasoning strategies that work best on ‘small but hard’ problems are ineffective on ‘big but easy’ problems and vice versa.     

PSL可满足问题的计算复杂度

PSL是一种用于描述并行系统的属性规约语言,包括线性时序逻辑FL和分支时序逻辑OBE两部分。由于OBE就是CTL,因此论文重点研究FL逻辑。理论上已证明许多难解的问题都可多项式变换为“可满足性”问题,“可满足性”问题是研究时序逻辑的核心问题之一,并已成为程序验证的一种有力工具;而计算复杂度是“可满足性”问题需要解决的最深刻的方向之一,其研究意义在于它可作为解决一类问题的难度的标准。文中在利用“铺砖模型”基础上,推导并得出FL的“可满足性”问题的计算复杂度为EXPSPACE—hard,这对正确评价解决该问题的各种算法的效率,进而确定对已有算法的改进余地具有重要的指导意义。