h-space structure in matrix displacement formulas

A class of spaces of matrices, calledh-spaces, is considered, extending previous results in [R. Bevilacqua, P. Zellini,Closure, commutativity and minmal complexity of some space of matrices, Linear and Multilinear Algebra,25, (1989) 1–25]. These spaces include several known classes of matrix algebras, such as group matrix algebras and Hessenberg algebras and, in particular, certain symmetric closed 1-spaces, which are structurally related to Toeplitz plus Hankel-like matrices. Following the displacement rank technique, these spaces are involved in general displacement decomposition formulas of an arbitrary matrixA. These decompositions lead to a significant representation formula for the inverse of a centrosymmetric Toeplitz plus Hankel matrix.     

Distributed SAT

We present DPLL ABT, a distributed Satisfiability solver (SAT) (Ansótegui and Manyà in IberoAm J Artif Intell 7(20):43–56, 2003) designed to solve distributed SAT problem instances. Since SAT is a particular case of constraint satisfaction, we propose a solving method based on the Asynchronous Backtracking algorithm (ABT) (Yokoo et al. in IEEE Trans Knowl Data Eng 10(5):673–685, 1998) developed for distributed constraint reasoning. In addition, we have applied the Davis-Putnam procedure (DPLL) in every agent, plus the minimum conflict heuristic in case DPLL does not detect any inconsistency. The resulting algorithm improves the performance in terms of communication cost and computational effort versus the basic ABT. The SAT instance is distributed into agents, which cooperate to solve SAT instances just sharing the minimum information. We also present the experimental results that demonstrate the performance of the method in terms of communication and execution time comparing the performance with the basic ABT algorithm.     

The SAT2002 competition

SAT Competition 2002 held in March–May 2002 in conjunction with SAT 2002 (the Fifth International Symposium on the Theory and Applications of Satisfiability Testing). About 30 solvers and 2300 benchmarks took part in the competition, which required more than 2 CPU years to complete the evaluation. In this report, we give the results of the competition, try to interpret them, and give suggestions for future competitions.     

Some structural properties of SAT

The following four conjectures about structural of SAT are studied in this paper.(1) SAT∈P^SPARSE∩NP;(2)SAT∈SRTDtt;(3)SAT∈Ptt^bAPP;(4)FPtt^SAT=FTlog^SAT.It is proved that some pairs of these conjectures imply P=NP ,for example,if SAT∈P^SPARSE∩NP and SAT∈Ptt^bAPP,or if SAT∈SRTDtt and SAT∩PttbAPP,then P=NP.This improves previous results in literature.     

The SAT2002 competition

SAT Competition 2002 held in March–May 2002 in conjunction with SAT 2002 (the Fifth International Symposium on the Theory and Applications of Satisfiability Testing). About 30 solvers and 2300 benchmarks took part in the competition, which required more than 2 CPU years to complete the evaluation. In this report, we give the results of the competition, try to interpret them, and give suggestions for future competitions.     

Extending the Noninterference Version of MLS for SAT

A noninterference formulation of MLS applicable to the Secure Ada® Target (SAT) Abstract Model is developed. An analogous formulation is developed to handle the SAT type enforcement policy. Unwinding theorems are presented for both MLS and Multidomain Security (MDS) and the SAT Abstract Model is shown to satisfy both MLS and MDS. Generalizations and extensions are also considered.     

Simple algorithm portfolio for SAT

The importance of algorithm portfolio techniques for SAT has long been noted, and a number of very successful systems have been devised, including the most successful one—SATzilla. However, all these systems are quite complex (to understand, reimplement, or modify). In this paper we present an algorithm portfolio for SAT that is extremely simple, but in the same time so efficient that it outperforms SATzilla. For a new SAT instance to be solved, our portfolio finds its k-nearest neighbors from the training set and invokes a solver that performs the best for those instances. The main distinguishing feature of our algorithm portfolio is the locality of the selection procedure—the selection of a SAT solver is based only on few instances similar to the input one. An open source tool that implements our approach is publicly available.     

高阶Hopfield神经网求解合取范式可满足性问题

本文提出用高阶Ｈｏｐｆｉｅｌｄ神经网络求解ＳＡＴ问题，给出了连续及离散高阶神经网络模型与相应的离散快速求解算法，证明了网络的稳定性，并用实验证明了该方法的可行性，且将该算法与Ｌｏｃａｌ Ｓｅａｒｃｈ算法进行了比较。     

Einstein谜的SAT求解

Einstein谜,亦称Zebra谜,是爱因斯坦在20世纪初提出的,他说世界上有98%的人答不出来。该问题是一个典型的逻辑推理题,可以通过SAT求解给出问题的答案。现将Einstein谜转换成SAT求解问题,并使用当前流行的SAT求解器,如MinSat,对Einstein谜进行自动求解。     

求解SAT问题的算法的研究进展

SAT问题是研究最广泛的NPC问题之一。由于SAT问题本身的特性,除非P=NP,否则不存在最坏情况下多项式阶时间复杂度的SAT求解算法。因此设计出高效快速的SAT求解算法至今仍是研究热点。首先简要介绍了SAT问题;其次从完备算法、不完备算法和组合算法3个角度总结了新近的研究进展,深入分析了已有算法解决SAT问题的基本流程,并从适用问题类别、算法特点、求解效率等方面对各类先进的求解器进行了对比分析;最后讨论了求解SAT问题的算法面临的挑战,并对下一步研究工作进行了展望。     

一个具有平均复杂性的SAT问题

最坏复杂性到平均复杂性的归约已被研究很多年。很多NP困难问题是最坏复杂性的。distNP类是平均复杂性的NP类,且有完全问题。LIVEN N证明了所有自然的NP完全问题都有平均复杂性的形式,但是他给出的概率分布是不自然的。在格问题方面,AJTAI和REGEV O分别提出了平均复杂性的困难问题,即短整数解问题(Short Integer Solution,SIS)和带错误学习问题(Learning With Errors,LWE)。给出一个可以归约到判定对于一个NP完全问题是否存在多项式时间算法,对值为1的实例给出一个见证,并且对值为0的实例给出一个归结辩驳的具有平均复杂性的SAT问题。本文方法是将求解NP完全问题的多项式时间算法的存在性转化成SAT问题的一组实例。同时也给出了这个问题的一些变形问题。     

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.     

Adaptive Application of SAT Solving Techniques

New heuristics and strategies have enabled major advancements in SAT solving in recent years. However, experimentation has shown that there is no winning solution that works in all cases. A degradation of orders of magnitude can be observed if the wrong heuristic is chosen. The problem is that it is impossible to know, in advance, which heuristics are best for a given problem. Consequently, many ideas - those that turn out to be useful for a small subset of the cases, but significantly increase run times on most others - are discarded.We propose the notion of Adaptive Solving as a possible solution to this problem. In our framework, the SAT solver monitors the effectiveness of the search on-the-fly using a Performance Metric. The metric gives a score according to its assessment of the search progress. Based on this score, one or more heuristics are turned on or off. The goal is to use a specific heuristic or strategy when it is advantageous, and turn it off when it is not, before it does too much damage. We suggest several possible metrics, and compare their effectiveness. Our adaptive solver achieves significant speedups on a large set of examples. We also show that applying different heuristics on different parts of the search space can improve run times even beyond what can be achieved by the best heuristic on its own.     

An overview of parallel SAT solving

Boolean satisfiability (SAT) solvers are currently very effective in practice. However, there are still many challenging problems for SAT solvers. Nowadays, extra computational power is no longer coming from higher processor frequencies. At the same time, multicore architectures are becoming predominant. Exploiting this new architecture is essential for the evolution of SAT solvers. Due to the increasing interest in parallel SAT solving, it is important to give an overview of what has been done so far. This paper presents an overview of parallel SAT solving and it is expected to be a valuable document for researchers in this field. This overview covers the main topics of parallel SAT solving, namely, different approaches and a variety of clause sharing strategies. Additionally, an evaluation of multicore SAT solvers is presented, showing the evolution of multicore SAT solvers over the last years.     

Visualizing SAT Instances and Runs of the DPLL Algorithm

SAT-solvers have turned into essential tools in many areas of applied logic like, for example, hardware verification or satisfiability checking modulo theories. However, although recent implementations are able to solve problems with hundreds of thousands of variables and millions of clauses, much smaller instances remain unsolved. What makes a particular instance hard or easy is at most partially understood – and is often attributed to the instance’s internal structure. By converting SAT instances into graphs and applying established graph layout techniques, this internal structure can be visualized and thus serve as the basis of subsequent analysis. Moreover, by providing tools that animate the structure during the run of a SAT algorithm, dynamic changes of the problem instance become observable. Thus, we expect both to gain new insights into the hardness of the SAT problem and to help in teaching SAT algorithms.     

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 (where x represents the lifetime) is the complement of the solvability function 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.     

A Stochastic Limit Approach to the SAT Problem

There exists an important problem whether there exists an algorithm to solve an NP-complete problem in polynomial time. In this paper, a new concept of quantum adaptive stochastic systems is proposed, and it is shown that it can be used to solve the problem above.     

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.     

Average-case analysis for the MAX-2SAT problem

We propose a simple probability model for MAX-2SAT instances for discussing the average-case complexity of the MAX-2SAT problem. Our model is a “planted solution model”, where each instance is generated randomly from a target solution. We show that for a large range of parameters, the planted solution (more precisely, one of the planted solution pairs) is the optimal solution for the generated instance with high probability. We then give a simple linear-time algorithm based on a message passing method, and we prove that it solves the MAX-2SAT problem with high probability for random MAX-2SAT instances under this planted solution model for probability parameters within a certain range.     

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.