Quantified maximum satisfiability

In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies an optimization extension of QBF and considers the problem in a quantified MaxSAT setting. More precisely, the general QMaxSAT problem is defined for QBFs with a set of soft clausal constraints and consists in finding the largest subset of the soft constraints such that the remaining QBF is true. Two approaches are investigated. One is based on relaxing the soft clauses and performing an iterative search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the two concrete problems of computing smallest minimal unsatisfiable subformulas (SMUS) and smallest minimal equivalent subformulas (SMES), decision versions of which are well-known problems in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the core-guided approach for the SMUS and SMES problems outperforms the use of iterative search over the values of the cost function. More significantly, the core-guided approach to SMUS also outperforms the state-of-the-art SMUS extractor Digger.     

On the performance of MaxSAT and MinSAT solvers on 2SAT-MaxOnes

We analyze and compare two solvers for Boolean optimization problems: WMaxSatz, a solver for Partial MaxSAT, and MinSatz, a solver for Partial MinSAT. Both MaxSAT and MinSAT are similar, but previous results indicate that when solving optimization problems using both solvers, the performance is quite different on some cases. For getting insights about the differences in the performance of the two solvers, we analyze their behaviour when solving 2SAT-MaxOnes problem instances, given that 2SAT-MaxOnes is probably the most simple, but NP-hard, optimization problem we can solve with them. The analysis is based first on the study of the bounds computed by both algorithms on some particular 2SAT-MaxOnes instances, characterized by the presence of certain particular structures. We find that the fraction of positive literals in the clauses is an important factor regarding the quality of the bounds computed by the algorithms. Then, we also study the importance of this factor on the typical case complexity of Random-p 2SAT-MaxOnes, a variant of the problem where instances are randomly generated with a probability p of having positive literals in the clauses. For the case p=0, the performance results indicate a clear advantage of MinSatz with respect to WMaxSatz, but as we consider positive values of p WMaxSatz starts to show a better performance, although at the same time the typical complexity of Random-p 2SAT-MaxOnes decreases as p increases. We also study the typical value of the bound computed by the two algorithms on these sets of instances, showing that the behaviour is consistent with our analysis of the bounds computed on the particular instances we studied first.     

基于不可满足原因的最小纠正集求解

布尔公式的最小纠正集MCS是子句的集合。对于一个不可满足公式,移除MCS后,所得到的新公式可满足。任一MCS中的子句保留在公式中,所得到的新公式不可满足。通过求解MCS 并调整约束集合,能够求解最小不可满足核心、MaxSAT 问题和最大（小）可满足解问题;还能够应用于故障定位、模型检查配置优化等实际问题中。 提出了一种基于不可满足原因的MCS求解算法,实现了相应的CUC工具。通过与目前最好的MCS求解工具LBX进行比较,得到了CUC性能优于LBX的结论。CUC比LBX平均多解出5%（65个）的公式。对于CUC和LBX均可解出的公式,CUC的平均求解时间比LBX快2.5倍。     

Resolution-based lower bounds in MaxSAT

The lower bound (LB) implemented in branch and bound MaxSAT solvers is decisive for obtaining a competitive solver. In modern solvers like MaxSatz and MiniMaxSat, the LB relies on the cooperation of the underestimation and inference components. Actually, the underestimation component of some solvers guides the application of the inference component when a conflict is reached and certain structures are found. In this paper we analyze algorithmic and logical aspects of the underestimation components that have been implemented in MaxSatz during its evolution. From an algorithmic point of view, we define novel strategies for selecting unit clauses in UP (the underestimation of LB in UP is the number of independent inconsistent subformulas detected using unit propagation), the extension of UP with failed literal detection, and a clever heuristic for guiding the application of MaxSAT resolution when UP augmented with failed literal detection is applied in the presence of cycles structures. From a logical point of view, we prove that the inconsistent subformulas detected by UP are minimally unsatisfiable, but this property does not hold if failed literal detection is added. In the presence of cycle structures in conflicts detected by UP augmented with failed literal detection, we prove that the application of MaxSAT resolution produces smaller inconsistent subformulas and, besides, generates additional clauses that may be used to improve the LB. The conducted empirical investigation indicates that the LB techniques described in this paper lead to better quality LBs.     

On computing minimal independent support and its applications to sampling and counting

Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in state-of-the-art constrained sampling and counting tools, while still retaining theoretical guarantees.     

一种求解布尔不可满足子式的局部搜索算法

解释布尔公式不可满足的原因在众多领域都具有非常重要的理论与应用价值,而不可满足子式能够为公式不可满足的原因提供精确的解释,帮助应用领域的自动化工具迅速定位错误,诊断问题失败的本质缘由。近年来涌现了许多基于SAT求解器DPLL回溯搜索过程的完全算法,但关于不完全方法提取不可满足子式的研究相对较少。因此,本文提出一种采用启发式局部搜索过程从公式的不可满足性证明中求解布尔不可满足子式的算法。该算法根据公式的消解规则通过局部搜索过程直接构造证明不可满足性的消解序列,并融合了布尔推理技术以提高搜索效率;而后通过一个递归过程遍历证明序列从而得到不可满足子式。通过实验与贪心遗传算法进行对比,结果表明本文提出的算法优于贪心遗传算法。     

A Parallelization Scheme Based on Work Stealing for a Class of SAT Solvers

Because of the inherent NP-completeness of SAT, many SAT problems currently cannot be solved in a reasonable time. Usually, in order to tackle a new class of SAT problems, new ad hoc algorithms must be designed. Another way to solve a new problem is to use a generic solver and employ parallelism to reduce the solve time. In this paper we propose a parallelization scheme for a class of SAT solvers based on the DPLL procedure. The scheme uses a dynamic load-balancing mechanism based on work-stealing techniques to deal with the irregularity of SAT problems. We parallelize Satz, one of the best generic SAT solvers, with our scheme to obtain a parallel solver called PSatz. The first experimental results on random 3-SAT problems and a set of well-known structured problems show the efficiency of PSatz. PSatz is freely available and runs on any networked workstations under Unix/Linux.     

More efficient two-mode stochastic local search for random 3-satisfiability

Stochastic local search (SLS) is a popular paradigm in incomplete solving for the Boolean satisfiability problem (SAT). Most SLS solvers for SAT switch between two modes, i.e., the greedy (intensification) mode and the diversification mode. However, the performance of these two-mode SLS algorithms lags far behind on solving random 3-satisfiability (3-SAT) problem, which is a significant special case of the SAT problem. In this paper, we propose a new hybrid scoring function called M C based on a linear combination of a greedy property m a k e and a diversification property C o n f T i m e s, and then utilize M C to develop a new two-mode SLS solver called CCMC. To evaluate the performance of CCMC, we conduct extensive experiments to compare CCMC with five state-of-the-art two-mode SLS solvers (i.e., Sparrow2011, Sattime2011, EagleUP, gNovelty+PCL and CCASat) on a broad range of random 3-SAT instances, including all large 3-SAT ones from SAT Competition 2009 and SAT Competition 2011 as well as 200 generated satisfiable huge random 3-SAT ones. The experiments illustrate that CCMC obviously outperforms its competitors, indicating the effectiveness of CCMC. We also analyze the effectiveness of the underlying ideas in CCMC and further improve the performance of CCMC on solving random 5-SAT instances.     

Propagation via lazy clause generation

Finite domain propagation solvers effectively represent the possible values of variables by a set of choices which can be naturally modelled as Boolean variables. In this paper we describe how to mimic a finite domain propagation engine, by mapping propagators into clauses in a SAT solver. This immediately results in strong nogoods for finite domain propagation. But a naive static translation is impractical except in limited cases. We show how to convert propagators to lazy clause generators for a SAT solver. The resulting system introduces flexibility in modelling since variables are modelled dually in the propagation engine and the SAT solver, and we explore various approaches to the dual modelling. We show that the resulting system solves many finite domain problems significantly faster than other techniques. This paper is an extension of results first published in [29, 30].     

Empirical hardness of finding optimal Bayesian network structures: algorithm selection and runtime prediction

Various algorithms have been proposed for finding a Bayesian network structure that is guaranteed to maximize a given scoring function. Implementations of state-of-the-art algorithms, solvers, for this Bayesian network structure learning problem rely on adaptive search strategies, such as branch-and-bound and integer linear programming techniques. Thus, the time requirements of the solvers are not well characterized by simple functions of the instance size. Furthermore, no single solver dominates the others in speed. Given a problem instance, it is thus a priori unclear which solver will perform best and how fast it will solve the instance. We show that for a given solver the hardness of a problem instance can be efficiently predicted based on a collection of non-trivial features which go beyond the basic parameters of instance size. Specifically, we train and test statistical models on empirical data, based on the largest evaluation of state-of-the-art exact solvers to date. We demonstrate that we can predict the runtimes to a reasonable degree of accuracy. These predictions enable effective selection of solvers that perform well in terms of runtimes on a particular instance. Thus, this work contributes a highly efficient portfolio solver that makes use of several individual solvers.     

Reducing the size of resolution proofs in linear time

DPLL-based SAT solvers progress by implicitly applying binary resolution. The resolution proofs that they generate are used, after the SAT solver’s run has terminated, for various purposes. Most notable uses in formal verification are: extracting an unsatisfiable core, extracting an interpolant, and detecting clauses that can be reused in an incremental satisfiability setting (the latter uses the proof only implicitly, during the run of the SAT solver). Making the resolution proof smaller can benefit all of these goals: it can lead to smaller cores, smaller interpolants, and smaller clauses that are propagated to the next SAT instance in an incremental setting. We suggest two methods that are linear in the size of the proof for doing so. Our first technique, called Recycle-Units uses each learned constant (unit clause) (x) for simplifying resolution steps in which x was the pivot, prior to when it was learned. Our second technique, called   simplifies proofs in which there are several nodes in the resolution graph, one of which dominates the others, that correspond to the same pivot. Our experiments with industrial instances show that these simplifications reduce the core by ≈5% and the proof by ≈13%. It reduces the core less than competing methods such as run- till- fix, but whereas our algorithms are linear in the size of the proof, the latter and other competing techniques are all exponential as they are based on SAT runs. If we consider the size of the proof (the resolution graph) as being polynomial in the number of variables (it is not necessarily the case in general), this gives our method an exponential time reduction comparing to existing tools for small core extraction. Our experiments show that this result is evident in practice more so for the second method: rarely it takes more than a few seconds, even when competing tools time out, and hence it can be used as a cheap proof post-processing procedure.     

Exploiting multivalued knowledge in variable selection heuristics for SAT solvers

We show that we can design and implement extremely efficient variable selection heuristics for SAT solvers by identifying, in Boolean clause databases, sets of Boolean variables that model the same multivalued variable and then exploiting that structural information. In particular, we define novel variable selection heuristics for two of the most competitive existing SAT solvers: Chaff, a solver based on look-back techniques, and Satz, a solver based on look-ahead techniques. Our heuristics give priority to Boolean variables that belong to sets of variables that model multivalued variables with minimum domain size in a given state of the search process. The empirical investigation conducted to evaluate the new heuristics provides experimental evidence that identifying multivalued knowledge in Boolean clause databases and using variable selection heuristics that exploit that knowledge leads to large performance improvements.      

Beyond the structure of SAT formulas

Nowadays, many real-world problems are encoded into SAT instances and efficiently solved by modern SAT solvers. These solvers, usually known as Conflict-Driven Clause Learning (CDCL) SAT solvers, include a variety of sophisticated techniques, such as clause learning, lazy data structures, conflict-based adaptive branching heuristics, or random restarts, among others. However, the reasons of their efficiency in solving real-world, or industrial, SAT instances are still unknown. The common wisdom in the SAT community is that these technique exploit some hidden structure of real-world problems.In this thesis, we characterize some important features of the underlying structure of industrial SAT instances. Namely, they are the community structure and the self-similar structure. We observe that most industrial SAT formulas, viewed as graphs, have these two properties. This means that (i) in a graph with a clear community structure, i.e. having high modularity, we can find a partition of its nodes into communities such that most edges connect nodes of the same community; and (ii) in a graph with a self-similar pattern, i.e. being fractal, its shape is kept after re-scalings, i.e., grouping sets of nodes into a single node. We also analyze how these structures are affected by the effects of CDCL techniques during the search.Using the previous structural studies, we propose three applications. First, we face the problem of generating pseudo-industrial random SAT instances using the notion of modularity. Our model generates instances similar to (classical) random SAT formulas when the modularity is low, but when this value is high, our model is also adequate to model realistic pseudo-industrial problems. Second, we propose a method based on the community structure of the instance to detect relevant learnt clauses. Our technique augments the original instance with this set of relevant clauses, and this results into an overall improvement of the efficiency of several state-of-the-art CDCL SAT solvers. Finally, we analyze the classification of industrial SAT instances into families using the previously analyzed structure features, and we compare them to other classifiers commonly used in portfolio SAT approaches.In summary, this dissertation extends the understandings of the structure of SAT instances, with the aim of better explaining the success of CDCL techniques and possibly improve them, and propose a number of applications based on this analysis of the underlying structure of SAT formulas.     

SMT proof checking using a logical framework

Producing and checking proofs from SMT solvers is currently the most feasible method for achieving high confidence in the correctness of solver results. The diversity of solvers and relative complexity of SMT over, say, SAT means that flexibility, as well as performance, is a critical characteristic of a proof-checking solution for SMT. This paper describes such a solution, based on a Logical Framework with Side Conditions (LFSC). We describe the framework and show how it can be applied for flexible proof production and checking for two different SMT solvers, clsat and cvc3. We also report empirical results showing good performance relative to solver execution time.     

NP-completeness of small conflict set generation for congruence closure

The efficiency of satisfiability modulo theories (SMT) solvers is dependent on the capability of theory reasoners to provide small conflict sets, i.e. small unsatisfiable subsets from unsatisfiable sets of literals. Decision procedures for uninterpreted symbols (i.e. congruence closure algorithms) date back from the very early days of SMT. Nevertheless, to the best of our knowledge, the complexity of generating smallest conflict sets for sets of literals with uninterpreted symbols and equalities had not yet been determined, although the corresponding decision problem was believed to be NP-complete. We provide here an NP-completeness proof, using a simple reduction from SAT.     

基于不可满足核的近似逼近可达性分析

近年来,形式化验证技术受到了越来越多的关注,它在保障安全关键领域系统的安全性和正确性方面发挥着重要的作用.模型检测作为形式化验证中自动化程度较高的分支,具有十分广阔的发展前景.本文中我们研究并提出了一种新的模型检测技术,可以有效地对迁移系统进行模型检测,包括不安全性检测和证明安全性.与现有的模型检测算法不同,我们提出的这种方法——基于不可满足核（unsatisfiable core,UC）的近似逼近可达性分析（UC-based approximate incremental reachability,UAIR）,主要利用不可满足核来求解一系列的候选安全不变式直至生成最终的不变式,以此来实现安全性证明和不安全性检测（漏洞查找）.在基于SAT求解器的符号模型检测中,我们使用由可满足性求解器得到的UC构造候选安全不变式,如果迁移系统本身是安全的,我们得到的初始不变式只是安全不变式的一个近似.然后,我们在检查安全性的同时,逐步改进候选安全不变式,直到找到一个真正的不变式,证明系统是安全的;如果系统是不安全的,我们的方法最终可以找到一个反例证明系统是不安全的.作为一种全新的方法,我们利用不可满足核进行安全性模型检测,取得了相当好的效果.众所周知,模型检测领域没有绝对最好的方法,尽管我们的方法在基准的可解数量上无法超越当前的成熟方法例如IC3、CAR等,但是我们的方法却可以解出3个其他方法都无法解出的案例,相信本方法可以作为模型检测工具集很有价值的补充.     

Don’t care in SMT: building flexible yet efficient abstraction/refinement solvers

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.