On exact selection of minimally unsatisfiable subformulae

A minimally unsatisfiable subformula (MUS) is a subset of clauses of a given CNF formula which is unsatisfiable but becomes satisfiable as soon as any of its clauses is removed. The selection of a MUS is of great relevance in many practical applications. This expecially holds when the propositional formula encoding the application is required to have a well-defined satisfiability property (either to be satisfiable or to be unsatisfiable). While selection of a MUS is a hard problem in general, we show classes of formulae where this problem can be solved efficiently. This is done by using a variant of Farkas' lemma and solving a linear programming problem. Successful results on real-world contradiction detection problems are presented.     

Generalizations of matched CNF formulas

A CNF formula is called matched if its associated bipartite graph (whose vertices are clauses and variables) has a matching that covers all clauses. Matched CNF formulas are satisfiable and can be recognized efficiently by matching algorithms. We generalize this concept and cover clauses by collections of bicliques (complete bipartite graphs). It turns out that such generalization indeed gives rise to larger classes of satisfiable CNF formulas which we term biclique satisfiable. We show, however, that the recognition of biclique satisfiable CNF formulas is NP-complete, and remains NP-hard if the size of bicliques is bounded. A satisfiable CNF formula is called var-satisfiable if it remains satisfiable under arbitrary replacement of literals by their complements. Var-satisfiable CNF formulas can be viewed as the best possible generalization of matched CNF formulas as every matched CNF formula and every biclique satisfiable CNF formula is var-satisfiable. We show that recognition of var-satisfiable CNF formulas is _P ² P-complete, answering a question posed by Kleine Büning and Zhao.     

Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference

A formula (in conjunctive normal form) is said to be minimal unsatisfiable if it is unsatisfiable and deleting any clause makes it satisfiable. The deficiency of a formula is the difference of the number of clauses and the number of variables. It is known that every minimal unsatisfiable formula has positive deficiency. Until recently, polynomial-time algorithms were known to recognize minimal unsatisfiable formulas with deficiency 1 and 2. We state an algorithm which recognizes minimal unsatisfiable formulas with any fixed deficiency in polynomial time.     

k-LSAT(k≥3)是NP-完全的(英文)

合取范式(conjunctive normal form,简称CNF)公式F是线性公式,如果F中任意两个不同子句至多有一个公共变元.如果F中的任意两个不同子句恰好含有一个公共变元,则称F是严格线性的.所有的严格线性公式均是可满足的,而对于线性公式类LCNF,对应的判定问题LSAT仍然是NP-完全的.LCNF≥k是子句长度大于或等于k的CNF公式子类,判定问题LSAT≥k的NP-完全性与LCNF≥k中是否含有不可满足公式密切相关.即LSAT≥k的NP-完全性取决于LCNF≥k是否含有不可满足公式.S.Porschen等人用超图和拉丁方的方法构造了LCNF≥3和LCNF≥4中的不可满足公式,并提出公开问题:对于k≥5,LCNF≥k是否含有不可满足公式?将极小不可满足公式应用于公式的归约,引入了一个简单的一般构造方法.证明了对于k≥3,k-LCNF含有不可满足公式,从而证明了一个更强的结果:对于k≥3,k-LSAT是NP-完全的.     

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical D^P-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999).Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails.     

针对MUS求解问题的加强剪枝策略

极小不可满足子集(minimal unsatisfiable subsets, MUS)的求解是布尔可满足性问题中的一个重要子问题. 对于一个给定的不可满足问题, 其MUS的求解能够反映出问题中导致其不可满足的关键原因. 然而, MUS的求解是一项极其耗时的任务, 不同的剪枝过程将直接影响到搜索空间的大小、算法的迭代次数, 从而影响算法的求解效率. 提出一种针对MUS求解的加强剪枝策略ABC (accelerating by critical MSS), 依据MSS、MCS、MUS这3者之间的对偶性和碰集关系特点, 提出cMSS和subMUS概念, 并总结出4条性质, 即每个MUS必是subMUS的超集, 进而在避免对MCS的碰集进行求解的情况下有效利用MUS和MCS互为碰集的特征, 有效避免求解碰集时的时间开销. 当subMUS不可满足时, 则subMUS是唯一的MUS, 算法将提前结束执行; 当subMUS可满足时, 则剪枝掉此节点, 进而有效避免对求解空间中的冗余空间进行搜索. 同时, 通过理论证明ABC策略的有效性, 并将其应用于目前最高效的单一化模型算法MARCO和双模型算法MARCO-MAM, 在标准测试用例下的实验结果表明, 该策略可以有效地对搜索空间进行进一步剪枝, 从而提高MUS的枚举效率.     

Generalizations of matched CNF formulas

A CNF formula is called matched if its associated bipartite graph (whose vertices are clauses and variables) has a matching that covers all clauses. Matched CNF formulas are satisfiable and can be recognized efficiently by matching algorithms. We generalize this concept and cover clauses by collections of bicliques (complete bipartite graphs). It turns out that such generalization indeed gives rise to larger classes of satisfiable CNF formulas which we term biclique satisfiable. We show, however, that the recognition of biclique satisfiable CNF formulas is NP-complete, and remains NP-hard if the size of bicliques is bounded. A satisfiable CNF formula is called var-satisfiable if it remains satisfiable under arbitrary replacement of literals by their complements. Var-satisfiable CNF formulas can be viewed as the best possible generalization of matched CNF formulas as every matched CNF formula and every biclique satisfiable CNF formula is var-satisfiable. We show that recognition of var-satisfiable CNF formulas is ₂ ^P-complete, answering a question posed by Kleine Büning and Zhao.     

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。     

A propositional theorem prover to solve planning and other problems

Classical STRIPS-style planning problems are formulated as theorems to be proven from a new point of view: that the problem is not solvable. The result for a refutation-based theorem prover may be a propositional formula that is to be proven unsatisfiable. This formula is identical to the formula that may be derived directly by various “SAT compilers”, but the theorem-proving view provides valuable additional information not in the formula, namely, the theorem to be proven. Traditional satisfiability methods, most of which are based on model search, are unable to exploit this additional information. However, a new algorithm called “Modoc” is able to exploit this information and has achieved performance comparable to the fastest known satisfiability methods, including stochastic search methods, on planning problems that have been reported by other researchers, as well as formulas derived from other applications. Unlike most theorem provers, Modoc performs well on both satisfiable and unsatisfiable formulas. Modoc works by a combination of back-chaining from the theorem clauses and forward-chaining on tractable subformulas. In some cases, Modoc is able to solve a planning problem without finding a complete assignment because the back-chaining methodology is able to ignore irrelevant clauses. Although back-chaining is well known in the literature, a high level of search redundancy existed in previous methods; Modoc incorporates a new technique called “autarky pruning”, which reduces search redundancy to manageable levels, permitting the benefits of back-chaining to emerge, for certain problem classes. Experimental results are presented for planning problems and formulas derived from other applications. This revised version was published online in June 2006 with corrections to the Cover Date.     

An approach for extracting a small unsatisfiable core

The article addresses the problem of finding a small unsatisfiable core of an unsatisfiable CNF formula. The proposed algorithm, CoreTrimmer, iterates over each internal node d in the resolution graph that ‘consumes’ a large number of clauses M (i.e., a large number of original clauses are present in the unsat core with the sole purpose of proving d) and attempts to prove them without the M clauses. If this is possible, it transforms the resolution graph into a new graph that does not have the M clauses at its core. CoreTrimmer can be integrated into a fixpoint framework similarly to Malik and Zhang’s fix-point algorithm run_till_ fix. We call this option trim_till_fix. Experimental evaluation on a large number of industrial CNF unsatisfiable formulas shows that trim_till_fix doubles, on average, the number of reduced clauses in comparison to run_till_fix. It is also better when used as a component in a bigger system that enforces short timeouts.     

不可满足公式的同态证明系统

合取范式(CNF)公式H到F的同态φ是一个从H的文字集合到F的文字集合的映射,并保持补运算和子句映到子句.同态映射保持一个公式的不可满足性.一个公式是极小不可满足的是指该公式本身不可满足,而且从中删去任意一个子句后得到的公式可满足.MU(1)是子句数与变元数的差等于1的极小不可满足公式类.一个三元组(H,φ,F)称为的一个来自H的同态证明,如果φ是一个从H到F的同态.利用基础矩阵的方法证明了:一个不可满足公式F的树消解证明,可以在多项式时间内转换成一个来自MU(1)中公式的同态证明.从而,由MU(1)中的公式构成的同态证明系统是完备的,并且由MU(1)中的公式构成的同态证明系统与树消解证明系统之间是多项式等价的.     

Random 2-SAT with Prescribed Literal Degrees

Two classic "phase transitions" in discrete mathematics are the emergence of a giant component in a random graph as the density of edges increases, and the transition of a random 2-SAT formula from satisfiable to unsatisfiable as the density of clauses increases. The random-graph result has been extended to the case of prescribed degree sequences, where the almost-sure nonexistence or existence of a giant component is related to a simple property of the degree sequence. We similarly extend the satisfiability result, by relating the almost-sure satisfiability or unsatisfiability of a random 2-SAT formula to an analogous property of its prescribed literal-degree sequence. The extension has proved useful in analyzing literal-degree-based algorithms for (uniform) random 3-SAT.     

Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints

Much research in the area of constraint processing has recently been focused on extracting small unsatisfiable “cores” from unsatisfiable constraint systems with the goal of finding minimal unsatisfiable subsets (MUSes). While most techniques have provided ways to find an approximation of an MUS (not necessarily minimal), we have developed a sound and complete algorithm for producing all MUSes of an unsatisfiable constraint system. In this paper, we describe a relationship between satisfiable and unsatisfiable subsets of constraints that we subsequently use as the foundation for MUS extraction algorithms, implemented for Boolean satisfiability constraints. The algorithms provide a framework with which many related subproblems can be solved, including relaxations of completeness to handle intractable instances, and we develop several variations of the basic algorithms to illustrate this. Experimental results demonstrate the performance of our algorithms, showing how the base algorithms run quickly on many instances, while the variations are valuable for producing results on instances whose complete results are intractably large. Furthermore, our algorithms are shown to perform better than the existing algorithms for solving either of the two distinct phases of our approach.     

An Efficient Approach to Solving Random k-sat Problems

Proving that a propositional formula is contradictory or unsatisfiable is a fundamental task in automated reasoning. This task is coNP-complete. Efficient algorithms are therefore needed when formulae are hard to solve. Random sat formulae provide a test-bed for algorithms because experiments that have become widely popular show clearly that these formulae are consistently difficult for any known algorithm. Moreover, the experiments show a critical value of the ratio of the number of clauses to the number of variables around which the formulae are the hardest on average. This critical value also corresponds to a ‘phase transition’ from solvability to unsolvability. The question of whether the formulae located around or above this critical value can efficiently be proved unsatisfiable on average (or even for a.e. formula) remains up to now one of the most challenging questions bearing on the design of new and more efficient algorithms. New insights into this question could indirectly benefit the solving of formulae coming from real-world problems, through a better understanding of some of the causes of problem hardness. In this paper we present a solving heuristic that we have developed, devoted essentially to proving the unsatisfiability of random sat formulae and inspired by recent work in statistical physics. Results of experiments with this heuristic and its evaluation in two recent sat competitions have shown a substantial jump in the efficiency of solving hard, unsatisfiable random sat formulae.     

Fast,flexible MUS enumeration

The problem of enumerating minimal unsatisfiable subsets (MUSes) of an infeasible constraint system is challenging due first to the complexity of computing even a single MUS and second to the potentially intractable number of MUSes an instance may contain. In the face of the latter issue, when complete enumeration is not feasible, a partial enumeration of MUSes can be valuable, ideally with a time cost for each MUS output no greater than that needed to extract a single MUS. Recently, two papers independently presented a new MUS enumeration algorithm well suited to partial MUS enumeration (Liffiton and Malik, 2013, Previti and Marques-Silva, 2013). The algorithm exhibits good anytime performance, steadily producing MUSes throughout its execution; it is constraint agnostic, applying equally well to any type of constraint system; and its flexible structure allows it to incorporate advances in single MUS extraction algorithms and eases the creation of further improvements and modifications. This paper unifies and expands upon the earlier work, presenting a detailed explanation of the algorithm’s operation in a framework that also enables clearer comparisons to previous approaches, and we present a new optimization of the algorithm as well. Expanded experimental results illustrate the algorithm’s improvement over past approaches and newly explore some of its variants.     

An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF

We consider the minimal unsatisfiability problem for propositional formulas over n variables with n+k clauses for fixedk. We will show that in case of at most n clauses no formula is minimal unsatisfiable. For n+1 clauses the minimal unsatisfiability problem is solvable in quadratic time. Further, we present a characterization of minimal unsatisfiable formulas with n+1clauses in terms of a certain form of matrices. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Complexity of Read-Once Resolution

We investigate the complexity of deciding whether a propositional formula has a read-once resolution proof. We give a new and general proof of Iwama–Miynano's theorem which states that the problem whether a formula has a read-once resolution proof is NP-complete. Moreover, we show for fixed k2 that the additional restriction that in each resolution step one of the parent clauses is a k-clause preserves the NP-completeness. If we demand that the formulas are minimal unsatisfiable and read-once refutable then the problem remains NP-complete. For the subclasses MU(k) of minimal unsatisfiable formulas we present a pol-time algorithm deciding whether a MU(k)-formula has a read-once resolution proof. Furthermore, we show that the problems whether a formula contains a MU(k)-subformula or a read-once refutable MU(k)-subformula are NP-complete.     

The satisfiability problem in regular CNF-formulas

 In this paper we deal with the propositional satisfiability (SAT) problem for a kind of multiple-valued clausal forms known as regular CNF-formulas and extend some known results from classical logic to this kind of formulas. We present a Davis–Putnam-style satisfiability checking procedure for regular CNF-formulas equipped with suitable data structures and prove its completeness. Then, we describe a series of experiments for regular random 3-SAT instances. We observe that, for the regular 3-SAT problem with this procedure, there exists a threshold of the ratio of clauses to variables such that (i) the most computationally difficult instances tend to be found near the threshold, (ii) there is a sharp transition from satisfiable to unsatisfiable instances at the threshold and (iii) the value of the threshold increases as the number of truth values considered increases. Instances in the hard part provide benchmarks for the evaluation of regular satisfiability solvers.     

A simplifier for propositional formulas with many binary clauses

Deciding whether a propositional formula in conjunctive normal form is satisfiable (SAT) is an NP-complete problem. The problem becomes linear when the formula contains binary clauses only. Interestingly, the reduction to SAT of a number of well-known and important problems--such as classical AI planning and automatic test pattern generation for circuits--yields formulas containing many binary clauses. In this paper we introduce and experiment with 2-SIMPLIFY, a formula simplifier targeted at such problems. 2-SIMPLIFY constructs the transitive closure of the implication graph corresponding to the binary clauses in the formula and uses this graph to deduce new unit literals. The deduced literals are used to simplify the formula and update the graph, and so on, until stabilization. Finally, we use the graph to construct an equivalent, simpler set of binary clauses. Experimental evaluation of this simplifier on a number of bench-mark formulas produced by encoding AI planning problems prove 2-SIMPLIFY to be a useful tool in many circumstances.     

基于寻找可满足2SAT子问题的SAT算法*

可满足问题（SAT）是一个NP-Hard问题。提出了一种求解SAT的新算法（FFSAT）。该算法将SAT问题转换为寻找一个可满足的2-SAT子问题。SAT问题虽然是NP完全问题,但是当所有子句长度不大于2时,SAT问题可以在线性时间求解。使用2-SAT算法-BinSat求解2-SAT子问题,当它不满足时,根据赋值选择新的2-SAT子问题。实验结果表明,采用本算法的结果优于UnitWalk。