Counting for Satisfiability by Inverting Resolution

We present a new algorithm for counting truth assignments of a clausal formula using inverse propositional resolution and its associated normalization rules. The idea is opposite of the classical resolution, and is achieved by constructing in a bottom-up manner a computation graph. This means that we successively add complementary literals to generate new bigger clauses instead of solving them. Next, we make a comparison between the classical and inverse resolution, followed by a new algorithm which combines these two techniques for solving the SAT problem.     

Solving satisfiability problems with preferences

Propositional satisfiability (SAT) is a success story in Computer Science and Artificial Intelligence: SAT solvers are currently used to solve problems in many different application domains, including planning and formal verification. The main reason for this success is that modern SAT solvers can successfully deal with problems having millions of variables. All these solvers are based on the Davis–Logemann–Loveland procedure (dll). In its original version, dll is a decision procedure, but it can be very easily modified in order to return one or all assignments satisfying the input set of clauses, assuming at least one exists. However, in many cases it is not enough to compute assignments satisfying all the input clauses: Indeed, the returned assignments have also to be “optimal” in some sense, e.g., they have to satisfy as many other constraints—expressed as preferences—as possible. In this paper we start with qualitative preferences on literals, defined as a partially ordered set (poset) of literals. Such a poset induces a poset on total assignments and leads to the definition of optimal model for a formula ψ as a minimal element of the poset on the models of ψ. We show (i) how dll can be extended in order to return one or all optimal models of ψ (once converted in clauses and assuming ψ is satisfiable), and (ii) how the same procedures can be used to compute optimal models wrt a qualitative preference on formulas and/or wrt a quantitative preference on literals or formulas. We implemented our ideas and we tested the resulting system on a variety of very challenging structured benchmarks. The results indicate that our implementation has comparable performances with other state-of-the-art systems, tailored for the specific problems we consider.     

Solving the resolution-free SAT problem by submodel propagation in linear time

We present a method, called Unicorn-SAT, based on submodel propagation, which solves the resolution-free SAT problem in linear time. A formula is resolution-free if there are no two clauses which differ only in one variable, i.e., each clause is blocked for each literal in it. A resolution-free formula is satisfiable or it contains the empty clause. For such a restricted formula we can find a model in linear time by submodel propagation. Submodel propagation is hyper-unit propagation by a submodel generated from a minimal clause. Hyper-unit propagation is unit propagation simultaneously by literals, as unit clauses, of a partial assignment. We obtain a submodel, i.e., a part of the model, by negation of a neighbor-resolution-mate of a minimal clause, which is a clause with the smallest number of literals in the formula. We obtain a neighbor-resolution-mate of a clause by negating one literal in it. By submodel propagation we obtain a formula which has fewer variables and clauses and remains resolution-free. Therefore, we can obtain a model by joining the submodels while we perform submodel propagation recursively until the formula becomes empty.     

SMELS: Satisfiability Modulo Equality with Lazy Superposition

We consider the problem of checking satisfiability of quantified formulae in First Order Logic with Equality. We propose a new procedure for combining SAT solvers with Superposition Theorem Provers to handle quantified formulae in an efficient and complete way. In our procedure, the input formula is converted into CNF as in traditional first order logic theorem provers. The ground clauses are given to the SAT solver, which runs a DPLL method to build partial models. The partial model is reduced, and then passed to a Superposition procedure, along with justifications of literals. The Superposition procedure then performs an inference rule, which we call Justified Superposition, between the ground literals and the nonground clauses, plus usual Superposition rules with the nonground clauses. Any resulting ground clauses are provided to the DPLL engine. We prove the completeness of our procedure, using a nontrivial modification of the Bachmair and Ganzinger’s model generation technique. We have implemented a theorem prover based on this idea by reusing state-of-the-art SAT solver and Superposition Theorem Prover. Our theorem prover inherits the best of both worlds: a SAT solver to handle ground clauses efficiently, and a Superposition theorem prover which uses powerful orderings to handle the nonground clauses. Experimental results are promising, and hereby confirm the viability of our method.     

Solving the resolution-free SAT problem by submodel propagation in linear time

We present a method, called Unicorn-SAT, based on submodel propagation, which solves the resolution-free SAT problem in linear time. A formula is resolution-free if there are no two clauses which differ only in one variable, i.e., each clause is blocked for each literal in it. A resolution-free formula is satisfiable or it contains the empty clause. For such a restricted formula we can find a model in linear time by submodel propagation. Submodel propagation is hyper-unit propagation by a submodel generated from a minimal clause. Hyper-unit propagation is unit propagation simultaneously by literals, as unit clauses, of a partial assignment. We obtain a submodel, i.e., a part of the model, by negation of a neighbor-resolution-mate of a minimal clause, which is a clause with the smallest number of literals in the formula. We obtain a neighbor-resolution-mate of a clause by negating one literal in it. By submodel propagation we obtain a formula which has fewer variables and clauses and remains resolution-free. Therefore, we can obtain a model by joining the submodels while we perform submodel propagation recursively until the formula becomes empty.Sponsored by Upper Austrian Government (Ph.D. scholarship) and SFB/FWF project P1302.     

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.     

多文字可满足SAT问题的相变点上界

可满足（SAT）问题是指：是否存在一组布尔变元赋值,使得随机合取范式（CNF）公式中每个子句至少有1个文字为真。多文字可满足SAT问题是指：是否存在一组布尔变元赋值,使得随机CNF公式中每个子句至少有2个文字为真。此问题仍然是一个NP难问题。定义约束密度α为CNF公式子句数与变元数之比,对该问题的相变点上界α*进行了研究。如果α>α*,则多文字可满足SAT问题高概率不可满足。通过一阶矩一个简单的推断,可以证明α*=-ln 2/ln(1-(k+1)/2k),当k=3时,α*=1。利用Kirousis等人的局部最大值技术,提升了多文字可满足3-SAT问题的相变点上界α*=0.7193。最后,选择了大量数据进行实验验证,结果表明,理论结果与实验结果相吻合。     

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。     

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.     

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.     

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.     

基于频次的SAT问题学习子句混合评估算法

为了有效管理学习子句,避免学习子句规模呈几何级增长,减少冗余学习子句对系统内存占用,从而提高布尔可满足性问题SAT求解器的求解效率,需要对学习子句进行评估,然后删减学习子句。传统的评估方式是基于学习子句的长度,保留较短的子句。当前主流的做法一个是变量衰减和VSIDS的子句评估方式,另外一个是基于文字块距离LBD的评估方式,也有将二者结合使用作为子句评估的依据。通过对学习子句参与冲突分析次数与问题求解的关系进行分析,将学习子句使用频率与LBD评估算法混合使用,既反映了学习子句在冲突分析中的作用,也充分利用了文字与决策层之间的信息。以Syrup求解器（GLUCOSE 4.1并行版本）为基准,在评估算法与并行子句共享策略方面做改进测试,通过实验对比发现,混合评估算法比LBD评估算法有优势,求解问题个数明显增多。     

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.     

Learning Conjunctions of Horn Clauses

An algorithm is presented for learning the class of Boolean formulas that are expressible as conjunctions of Horn clauses. (A Horn clause is a disjunction of literals, all but at most one of which is a negated variable.) The algorithm uses equivalence queries and membership queries to produce a formula that is logically equivalent to the unknown formula to be learned. The amount of time used by the algorithm is polynomial in the number of variables and the number of clauses in the unknown formula.     

Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses

We prove hardness results for approximating set splitting problems andalso instances of satisfiability problems which have no “mixed”clauses, i.e., every clause has either all its literals unnegated orall of them negated. Results of Håstad imply tighthardness results for set splitting when all sets have size exactly $k\ge 4$ elements and also for non-mixed satisfiability problems withexactly $k$ literals in each clause for $k \ge 4$. We consider thecase $k=3$. For the MAX E3-SET SPLITTING, problem in which all sets have sizeexactly 3, we prove an NP-hardness result for approximating withinany factor better than ${\frac{19}{20}}$. This result holds even for satisfiableinstances of MAX E3-SET SPLITTING, and is based on a PCP construction due toHåstad. For “non-mixed MAX 3SAT,” we give aPCP construction which is a slight variant of the one given by Håstad for MAX 3SAT, and use it to prove the NP-hardness ofapproximating within a factor better than ${\frac{11}{12}}$.     

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.     

Toward leaner binary-clause reasoning in a satisfiability solver

Binary-clause reasoning has been shown to reduce the size of the search space on many satisfiability problems, but has often been so expensive that run-time was higher than that of a simpler procedure that explored a larger space. The method of Sharir for detecting strongly connected components in a directed graph can be adapted to performing lean resolution on a set of binary clauses. Beyond simply detecting unsatisfiability, the goal is to find implied equivalent literals, implied unit clauses, and implied binary clauses.     

A Taxonomy of Exact Methods for Partial Max-SAT

Partial Maximum Boolean Satisfiability (Partial Max-SAT or PMSAT) is an optimization variant of Boolean satisfiability (SAT) problem, in which a variable assignment is required to satisfy all hard clauses and a maximum number of soft clauses in a Boolean formula. PMSAT is considered as an interesting encoding domain to many real-life problems for which a solution is acceptable even if some constraints are violated. Amongst the problems that can be formulated as such are planning and scheduling. New insights into the study of PMSAT problem have been gained since the introduction of the Max-SAT evaluations in 2006. Indeed, several PMSAT exact solvers have been developed based mainly on the Davis-Putnam-Logemann-Loveland (DPLL) procedure and Branch and Bound (B&B) algorithms. In this paper, we investigate and analyze a number of exact methods for PMSAT. We propose a taxonomy of the main exact methods within a general framework that integrates their various techniques into a unified perspective. We show its effectiveness by using it to classify PMSAT exact solvers which participated in the 2007～2011 Max-SAT evaluations, emphasizing on the most promising research directions.     

An Effective Algorithm for the Futile Questioning Problem

In the futile questioning problem, one must decide whether acquisition of additional information can possibly lead to the proof of a conclusion. Solution of that problem demands evaluation of a quantified Boolean formula at the second level of the polynomial hierarchy. The same evaluation problem, called Q-ALL SAT, arises in many other applications. In this paper, we introduce a special subclass of Q-ALL SAT that is at the first level of the polynomial hierarchy. We develop a solution algorithm for the general case that uses a backtracking search and a new form of learning of clauses. Results are reported for two sets of instances involving a robot route problem and a game problem. For these instances, the algorithm is substantially faster than state-of-the-art solvers for quantified Boolean formulas.     

Toward leaner binary-clause reasoning in a satisfiability solver

Binary-clause reasoning has been shown to reduce the size of the search space on many satisfiability problems, but has often been so expensive that run-time was higher than that of a simpler procedure that explored a larger space. The method of Sharir for detecting strongly connected components in a directed graph can be adapted to performing lean resolution on a set of binary clauses. Beyond simply detecting unsatisfiability, the goal is to find implied equivalent literals, implied unit clauses, and implied binary clauses.