Clause Weighting Local Search for SAT

This paper investigates the necessary features of an effective clause weighting local search algorithm for propositional satisfiability testing. Using the recent history of clause weighting as evidence, we suggest that the best current algorithms have each discovered the same basic framework, that is, to increase weights on false clauses in local minima and then to periodically normalize these weights using a decay mechanism. Within this framework, we identify two basic classes of algorithm according to whether clause weight updates are performed additively or multiplicatively. Using a state-of-the-art multiplicative algorithm (SAPS) and our own pure additive weighting scheme (PAWS), we constructed an experimental study to isolate the effects of multiplicative in comparison to additive weighting, while controlling other key features of the two approaches, namely, the use of pure versus flat random moves, deterministic versus probabilistic weight smoothing and multiple versus single inclusion of literals in the local search neighbourhood. In addition, we examined the effects of adding a threshold feature to multiplicative weighting that makes it indifferent to similar cost moves. As a result of this investigation, we show that additive weighting can outperform multiplicative weighting on a range of difficult problems, while requiring considerably less effort in terms of parameter tuning. Our examination of the differences between SAPS and PAWS suggests that additive weighting does benefit from the random flat move and deterministic smoothing heuristics, whereas multiplicative weighting would benefit from a deterministic/probabilistic smoothing switch parameter that is set according to the problem instance. We further show that adding a threshold to multiplicative weighting produces a general deterioration in performance, contradicting our earlier conjecture that additive weighting has better performance due to having a larger selection of possible moves. This leads us to explain differences in performance as being mainly caused by the greater emphasis of additive weighting on penalizing clauses with relatively less weight.     

采用二元CSP引擎求解RTL数据通路的可满足性

针对寄存器传输级(RTL)验证和测试过程中非常重要的数据通路可满足性求解问题,提出一种基于二元约束满足问题(CSP)的求解方法,包括数据通路提取、二元CSP建模和搜索求解3个步骤.数据通路提取通过对接口布尔变量和某些字变量赋值,为各个数据通路器件建立环境;二元CSP建模则根据该环境和各个数据通路器件的功能,将数据通路的可满足性问题转化为二元CSP描述;该二元CSP问题的描述被送入到二元CSP引擎,并采用冲突引导的回跳搜索策略进行求解,获得有解的例证或无解的判定.实验结果表明,即使在没有采取很多优化策略的条件下,该方法仍有较好的性能,并优于基于线性规划(LP)的求解方法.     

Compiling finite linear CSP into SAT

In this paper, we propose a new method to encode Constraint Satisfaction Problems (CSP) and Constraint Optimization Problems (COP) with integer linear constraints into Boolean Satisfiability Testing Problems (SAT). The encoding method (named order encoding) is basically the same as the one used to encode Job-Shop Scheduling Problems by Crawford and Baker. Comparison x ≤ a is encoded by a different Boolean variable for each integer variable x and integer value a. To evaluate the effectiveness of this approach, we applied the method to the Open-Shop Scheduling Problems (OSS). All 192 instances in three OSS benchmark sets are examined, and our program found and proved the optimal results for all instances including three previously undecided problems.     

Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems

We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or heavy tails. We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavy-tailed behavior. Furthermore, for harder problem instances, we observe long tails on the left-hand side of the distribution, which is indicative of a non-negligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavy-tailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis.     

可满足性问题的研究综述

对SAT问题及其各种约束子问题进行分类并给出具体定义,着重介绍常规SAT问题、最大可满足性问题（MAX—SAT）和参数化SAT问题的相关算法,并对参数算法中运用的技术进行分析和比较,提出一些SAT问题研究中值得关注的几个方面。     

A SAT Solver Using Reconfigurable Hardware and Virtual Logic

In this paper, we present the architecture of a new SAT solver using reconfigurable logic and a virtual logic scheme. Our main contributions include new forms of massive fine-grain parallelism, structured design techniques based on iterative logic arrays that reduce compilation times from hours to minutes, and a decomposition technique that creates independent subproblems that may be concurrently solved by unconnected FPGAs. The decomposition technique is the basis of the virtual logic scheme, since it allows solving problems that exceed the hardware capacity. Our architecture is easily scalable. Our results show several orders of magnitude speedup compared with a state-of-the-art software implementation, and also with respect to prior SAT solvers using reconfigurable hardware.     

A parsimony tree for the SAT2002 competition

Twenty of the programs (solvers) submitted to the SAT 2002 Contest had no disqualifying errors. These solvers were run on 2023 satisfiability problems of varying hardnesses. Each solver was judged by which problems it could solve within the allowed time limit. Twelve solvers were best on some problem – they could solve it and the others could not. Only two solvers could not beat each remaining solver on some problems (where the problems could vary depending on which solver it was trying to beat). Thus, there is evidence that 18 solvers were extremely good. It is interesting to analyze the contest results in a way that groups together solvers with similar strengths and weaknesses. This paper uses the parsimony algorithm to produce a classification of the twenty solvers. The paper also has a second classification, almost the same as the first, for the twenty solvers, updated versions of two solvers, and a fictitious state of the art solver. The contest problems came in three groups, Industrial, Hand Made, and Random. The Random group of problems was about three times as large as the other two together. The classification identifies four groups of solvers (plus a miscellaneous group): weak solvers, incomplete solvers which are very good at some satisfiable Random problems, complete solvers which are very good at most Random problems, and complete solvers which are very good at Industrial and Hand Made problems.     

A parsimony tree for the SAT2002 competition

Twenty of the programs (solvers) submitted to the SAT 2002 Contest had no disqualifying errors. These solvers were run on 2023 satisfiability problems of varying hardnesses. Each solver was judged by which problems it could solve within the allowed time limit. Twelve solvers were best on some problem — they could solve it and the others could not. Only two solvers could not beat each remaining solver on some problems (where the problems could vary depending on which solver it was trying to beat). Thus, there is evidence that 18 solvers were extremely good. It is interesting to analyze the contest results in a way that groups together solvers with similar strengths and weaknesses. This paper uses the parsimony algorithm to produce a classification of the twenty solvers. The paper also has a second classification, almost the same as the first, for the twenty solvers, updated versions of two solvers, and a fictitious state of the art solver. The contest problems came in three groups, Industrial, Hand Made, and Random. The Random group of problems was about three times as large as the other two together. The classification identifies four groups of solvers (plus a miscellaneous group): weak solvers, incomplete solvers which are very good at some satisfiable Random problems, complete solvers which are very good at most Random problems, and complete solvers which are very good at Industrial and Hand Made problems.     

Combining SAT Methods with Non-Clausal Decision Heuristics

A decision procedure for arbitrary first-order formulas can be viewed as combining a propositional search with a decision procedure for conjunctions of first-order literals, so Boolean SAT methods can be used for the propositional search in order to improve the performance of the overall decision procedure. We show how to combine some Boolean SAT methods with non-clausal heuristics developed for first-order decision procedures. The combination of methods leads to a smaller number of decisions than either method alone.     

Efficient data structures for backtrack search SAT solvers

The implementation of efficient Propositional Satisfiability (SAT) solvers entails the utilization of highly efficient data structures, as illustrated by most of the recent state-of-the-art SAT solvers. However, it is in general hard to compare existing data structures, since different solvers are often characterized by fairly different algorithmic organizations and techniques, and by different search strategies and heuristics. This paper aims the evaluation of data structures for backtrack search SAT solvers, under a common unbiased SAT framework. In addition, advantages and drawbacks of each existing data structure are identified. Finally, new data structures are proposed, that are competitive with the most efficient data structures currently available, and that may be preferable for the next generation SAT solvers.     

求解加热炉调度的改进修复式约束满足算法

针对钢铁生产中加热炉调度问题,考虑炉容受限的情况,以最小化板坯的Makespan和最小化总在炉加工时间为目标建立问题的多目标优化模型,将其归结为多旅行商问题。针对问题的NP-难特性,提出一种改进的修复式约束满足算法求解。松弛炉容约束得到初始调度,在检测冲突变量并构造冲突板坯的可替换加热炉集合的基础上,以开工时间偏移最小规则为冲突板坯重新指派加热炉,得到可行的调度方案。数据实验验证了模型和算法的可行性和有效性。     

Efficient data structures for backtrack search SAT solvers

The implementation of efficient Propositional Satisfiability (SAT) solvers entails the utilization of highly efficient data structures, as illustrated by most of the recent state-of-the-art SAT solvers. However, it is in general hard to compare existing data structures, since different solvers are often characterized by fairly different algorithmic organizations and techniques, and by different search strategies and heuristics. This paper aims the evaluation of data structures for backtrack search SAT solvers, under a common unbiased SAT framework. In addition, advantages and drawbacks of each existing data structure are identified. Finally, new data structures are proposed, that are competitive with the most efficient data structures currently available, and that may be preferable for the next generation SAT solvers.     

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.     

Best Possible Approximation Algorithm for MAX SAT with Cardinality Constraint

We consider the MAX SAT problem with the additional constraint that at most P variables have a true value. We obtain a (1-e ^-1 ) -approximation algorithm for this problem. Feige [6] has proved that for MAX SAT with cardinality constraint with clauses without negations this is the best possible performance guarantee unless P=NP . Received August 20, 1998; revised June 23, 1999, and April 17, 2000.     

A Methodological View of Constraint Solving

Constraints are an effective tool to define sets of data by means of logical formulae. Our goal here is to survey the notion of constraint system and to give examples of constraint systems operating on various domains, such as natural, rational or real numbers, finite domains, and term domains. We classify the different methods used for solving constraints, syntactic methods based on transformations, semantic methods based on adequate representations of constraints, hybrid methods combining transformations and enumerations. The concepts and methods are illustrated via examples. We also discuss applications of constraints to various fields, such as programming, operations research, and theorem proving.     

UnitWalk: A new SAT solver that uses local search guided by unit clause elimination

In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification. Our algorithm is inspired by two randomized algorithms having the best current worst-case upper bounds ([27,28] and [30,31]). We combine the main ideas of these algorithms in one algorithm. The two approaches we use are local search (which is used in many SAT algorithms, e.g., in GSAT [34] and WalkSAT [33]) and unit clause elimination (which is rarely used in local search algorithms). In this paper we do not prove any theoretical bounds. However, we present encouraging results of computational experiments comparing several implementations of our algorithm with other SAT solvers. We also prove that our algorithm is probabilistically approximately complete (PAC).     

UnitWalk: A new SAT solver that uses local search guided by unit clause elimination

In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification. Our algorithm is inspired by two randomized algorithms having the best current worst-case upper bounds ([27,28] and [30,31]). We combine the main ideas of these algorithms in one algorithm. The two approaches we use are local search (which is used in many SAT algorithms, e.g., in GSAT [34] and WalkSAT [33]) and unit clause elimination (which is rarely used in local search algorithms). In this paper we do not prove any theoretical bounds. However, we present encouraging results of computational experiments comparing several implementations of our algorithm with other SAT solvers. We also prove that our algorithm is probabilistically approximately complete (PAC).     

On SAT instance classes and a method for reliable performance experiments with SAT solvers

A recent series of experiments with a group of state-of-the-art SAT solvers and several well-defined classes of problem instances reports statistically significant performance variability for the solvers. A systematic analysis of the observed performance data, all openly archived on the Web, reveals distributions which we classify into three broad categories: (1) readily characterized with a simple ²-test, (2) requiring more in-depth analysis by a statistician, (3) incomplete, due to time-out limit reached by specific solvers. The first category includes two well-known distributions: normal and exponential; we use simple first-order criteria to decide the second category and label the distributions as near-normal, near-exponential and heavy-tail. We expect that good models for some if not most of these may be found with parameters that fit either generalized gamma, Weibull, or Pareto distributions. Our experiments show that most SAT solvers exhibit either normal or exponential distribution of execution time (runtime) on many equivalence classes of problem instances. This finding suggests that the basic mathematical framework for these experiments may well be the same as the one used to test the reliability or lifetime of hardware components such as lightbulbs, A/C units, etc. A batch of N replicated hardware components represents an equivalence class of N problem instances in SAT, a controlled operating environment A represents a SAT solver A, and the survival function R ^A (x) (where x represents the lifetime) is the complement of the solvability function S ^A (x)=1–R ^A (x) where x may represent runtime, implications, backtracks, etc. As demonstrated in the paper, a set of unrelated benchmarks or randomly generated SAT instances available today cannot measure the performance of SAT solvers reliably – there is no control on their 'hardness'. However, equivalence class instances as defined in this paper are, in effect, replicated instances of a specific reference instance. The proposed method not only provides a common platform for a systematic study and a reliable improvement of deterministic and stochastic SAT solvers alike but also supports the introduction and validation of new problem instance classes.     

Solution Techniques for Constraint Satisfaction Problems: Advanced Approaches

Conventional techniques for the constraint satisfaction problem (CSP)have had considerable success in their applications. However,there are many areas in which the performance of the basic approachesmay be improved. These include heuristic ordering of certain tasksperformed by the CSP solver, hybrids which combine compatible solutiontechniques and graph based methods which exploit the structure of theconstraint graph representation of a CSP. Also, conventionalconstraint satisfaction techniques only address problems with hardconstraints (i.e. each of which are completely satisfied or completelyviolated, and all of which must be satisfied by a validsolution). Many real applications require a more flexible approachwhich relaxes somewhat these rigid requirements. To address theseissues various approaches have been developed. This paper attempts asystematic review of them.     

可满足问题中的模型计数

模型计数问题是指计算给定问题的解的个数,这是一类比决策更困难的问题,也是人工智能领域研究的一个热点问题.对模型计数问题的研究不仅可以提高算法的求解效率,更能促进对问题困难本质的了解.以可满足问题（命题可满足（SAT）和约束可满足问题（CSP））为例,从精确算法和近似求解两方面综述了模型计数问题的研究现状,重点介绍了相关概念以及各个算法之间的优缺点,并提出了有待解决的开放性问题,对模型计数问题的研究予以了总结和展望.