Computing Maximal Autarkies with Few and Simple Oracle Queries

The topic of this thesis is the development of SAT-based techniques and tools for solving industrial combinatorial problems. First, it describes the architecture of state-of-the-art SAT and SMT Solvers based on the classical DPLL procedure. These systems can be used as black boxes for solving combinatorial problems. However, sometimes we can increase their efficiency with slight modifications of the basic algorithm. Therefore, the study and development of techniques for adjusting SAT Solvers to specific combinatorial problems is the first goal of this thesis. Namely, SAT Solvers can only deal with propositional logic. For solving general combinatorial problems, two different approaches are possible: - Reducing the complex constraints into propositional clauses. - Enriching the SAT Solver language. The first approach corresponds to encoding the constraint into SAT. The second one corresponds to using propagators, the basis for SMT Solvers. Regarding the first approach, in this document we improve the encoding of two of the most important combinatorial constraints: cardinality constraints and pseudo-Boolean constraints. After that, we present a new mixed approach, called lazy decomposition, which combines the advantages of encodings and propagators. The other part of the thesis uses these theoretical improvements in industrial combinatorial problems. We give a method for efficiently scheduling some professional sport leagues with SAT. The results are promising and show that a SAT approach is valid for these problems. However, the chaotical behavior of CDCL-based SAT Solvers due to VSIDS heuristics makes it difficult to obtain a similar solution for two similar problems. This may be inconvenient in real-world problems, since a user expects similar solutions when it makes slight modifications to the problem specification. In order to overcome this limitation, we have studied and solved the close solution problem, i.e., the problem of quickly finding a close solution when a similar problem is considered.

In this work, we improve on existing work that studied the relationship between the proof system of modern SAT solvers and general resolution. Previous contributions such as those by Beame et al (2004), Hertel et al (2008), and Buss et al (2008) demonstrated that variations on modern clause-learning SAT solvers were as powerful as general resolution. However, the models used in these studies required either extra degrees of non-determinism or a preprocessing step that are not utilized by any state-of-the-art SAT solvers in practice. In this paper, we prove that modern SAT solvers that learn asserting clauses indeed p-simulate general resolution without the need for any additional techniques.

This paper continued the study of the complexity behavior of the satisfiability analysis of hard random formulas given in the conjunctive normal form (CNF). In the 3-CNF formula, each clause contains three literals of logical variables. The number of logical variables in the formula is N. In this paper, the SAT solver is improved by introducing equality reduction and pure literal identification procedures. The solver improvement reduced the exponent (with base 2) from N/20.86 to N/21.41 with an R=4.6 ratio of the number of clauses to the number of variables. The results show that the efficiency of the pure literals identification procedure decreases as R increases. An important part of the study is an empirical estimation of the algorithmic complexity of the SAT problem with large number of variables. The proposed method gives a convenient lower bound on the complexity of analysis for random 3-CNF formulas. We estimated the algorithmic complexity for the range N=256÷8192. The exponential dependence of complexity on N for random 3-CNF formulas at a fixed value of R is demonstrated in this range.

Maximum Satisfiability (MaxSAT) is an optimization variant of the Boolean Satisfiability (SAT) problem. In general, MaxSAT algorithms perform a succession of SAT solver calls to reach an optimum solution making extensive use of cardinality constraints. Many of these algorithms are non-incremental in nature, i.e. at each iteration the formula is rebuilt and no knowledge is reused from one iteration to another. In this paper, we exploit the knowledge acquired across iterations using novel schemes to use cardinality constraints in an incremental fashion. We integrate these schemes with several MaxSAT algorithms. Our experimental results show a significant performance boost for these algorithms as compared to their non-incremental counterparts. These results suggest that incremental cardinality constraints could be beneficial for other constraint solving domains.

Software product lines are able to describe multiple products sharing a common base of features and are commonly described as feature models. For complex software product lines, automatic analyses are required to ensure validity and to improve the interactive configuration process. Modern SAT solvers are vital components for the validation process of feature models. The increasing variability of software product lines implies the need to use more expressive solvers like SMT solvers. To assist the development of feature modeling tools, we compare SAT and SMT solvers for the automated analysis of feature models. During this thesis, we create an abstract data type to formally define analyses for feature model defects and their explanations. The result shows that SAT solvers are more efficient at detecting the defects, while SMT solvers can find explanations for them multiple times faster. Feature models can be further expanded by attaching attributes to features. Such attributes may contain a numerical value. Additionally, one attribute can be defined for multiple features. In this thesis, we aim to support the interactive configuration process, by providing the range of the sum of values for an attribute. Such ranges depend on the remaining choices in a configuration of the product line. We provide an exact computation using SMT and an approximation using a heuristic. The evaluation results show that an SMT solver is not suitable for supporting interactive configuration. However, the approximated ranges of the provided heuristic were very close to the exact ones.

Answer Set Programming (ASP) is a well-known declarative problem solving approach based on nonmonotonic logic programs, which has been successfully applied to a wide range of applications in artificial intelligence and beyond. To address the needs of modern applications, HEX-programs were introduced as an extension of ASP with external atoms for accessing information outside programs via an API style bi-directional interface mechanism. To evaluate such programs, conflict-driving learning algorithms for SAT and ASP solving have been extended in order to capture the semantics of external atoms. However, a drawback of the state-of-the-art approach is that external atoms are only evaluated under complete assignments (i.e., input to the external source) while in practice, their values often can be determined already based on partial assignments alone (i.e., from incomplete input to the external source). This prevents early backtracking in case of conflicts, and hinders more efficient evaluation of HEX-programs. We thus extend the notion of external atoms to allow for three-valued evaluation under partial assignments, while the two-valued semantics of the overall HEX-formalism remains unchanged. This paves the way for three enhancements: first, to evaluate external sources at any point during model search, which can trigger learning knowledge about the source behavior and/or early backtracking in the spirit of theory propagation in SAT modulo theories (SMT). Second, to optimize the knowledge learned in terms of so-called nogoods, which roughly speaking are impossible input-output configurations. Shrinking nogoods to their relevant input part leads to more effective search space pruning. And third, to make a necessary minimality check of candidate answer sets more efficient by exploiting early external evaluation calls. As this check usually accounts for a large share of the total runtime, optimization is here particularly important. We further present an experimental evaluation of an implementation of a novel HEX-algorithm that incorporates these enhancements using a benchmark suite. Our results demonstrate a clear efficiency gain over the state-of-the-art HEX-solver for the benchmarks, and provide insights regarding the most effective combinations of solver configurations.

When a circuit exhibits abnormal behaviour, a diagnosis of the circuit is a set of gates whose failure explains the abnormality. The cardinality of a diagnosis is the number of gates assumed to be failing. A diagnosis is of minimum cardinality if no other diagnosis of the circuit exists that has a smaller cardinality. In general, the number of diagnoses can be exponentially large, and often only a preferred set of minimum-cardinality diagnoses is computed. In propositional satisfiability (SAT), given a propositional circuit, the task is to check whether all gates of the circuit can be assigned values in a consistent manner. In SAT-based diagnosis approach, the given circuit is injected with additional circuitry modelling the health of gates as well as the cardinality constraint. Under an abnormal observation the SAT solver repeatedly computes consistent assignments to all gates in the augmented circuit where each such assignment corresponds to a minimum-cardinality diagnosis. However, diagnosis of large size circuits as well as diagnostic cases with large minimum cardinalities pose a challenge for SAT solvers. To scale diagnosis to larger and harder cases, we propose a novel encoding that captures the hierarchical structure of the circuit in terms of single-output self-contained sub-circuits called cones. Cones have been exploited in diagnostic reasoning in the past; however, our encoding is the first of its kind in a SAT-based approach. Previously, cones were used to take the abstraction of a circuit and simplify it, while we exploit cones to improve the speed and efficiency of the SAT solver. Experiments on 1800 diagnostic cases of ISCAS-85 benchmark circuits show that the new encoding allows faster and more scalable diagnosis solving 1700 cases which is 163 more than the number of cases solved by the baseline approach.

We formalize the class of “sequential local and show that these algorithms fail to find satisfying assignments on random instances of the “Not-All-Equal-$K$-SAT” (NAE-$K$-SAT) problem if the number of message passing iterations is bounded by a function moderately growing in the number of variables and if the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ for sufficiently large $K$. Sequential local algorithms are those that iteratively set variables based on some local information and/or local randomness and then recurse on the reduced instance. Our model captures some weak abstractions of natural algorithms such as Survey Propagation (SP)-guided as well as Belief Propagation (BP)-guided decimation algorithms---two widely studied message-passing--based algorithms---when the number of message-passing rounds in these algorithms is restricted to be growing only moderately with the number of variables. The approach underlying our paper is based on an intricate geometry of th...

The Boolean Satisfiability Problem (SAT) is well known in computational complexity, representing the first decision problem to be proved NP-complete. SAT is also often the subject of work in proof complexity. Besides its theoretical interest, SAT finds a wide range of practical applications. Moreover, SAT solvers have been the subject of remarkable efficiency improvements since the mid-1990s, motivating their widespread use in many practical applications including Bounded and Unbounded Model Checking. The success of SAT solvers has also motivated the development of algorithms for natural extensions of SAT, including Quantified Boolean Formulas (QBF), Pseudo-Boolean constraints (PB), Maximum Satisfiability (MaxSAT) and Satisfiability Modulo Theories (SMT) which also see application in the model-checking context. This chapter first covers the organization of modern conflict-driven clause learning (CDCL) SAT solvers, which are used in the vast majority of practical applications of SAT. It then reviews the techniques shown to be effective in modern SAT solvers.

Given a Boolean formula $\phi$ over $n$ variables, the problem of model counting is to compute the number of solutions of $\phi$. Model counting is a fundamental problem in computer science with wide-ranging applications. Owing to the \#P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that $\log n$ calls to an NP oracle are necessary and sufficient to achieve $(\varepsilon,\delta)$ guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting. The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide $n$ bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

Linear search algorithms have been shown to be particularly effective for solving partial Maximum Satisfiability (MaxSAT) problem instances. These algorithms start by adding a new relaxation variable to each soft clause and solving the resulting formula with a SAT solver. Whenever a Model is found, a new constraint on the relaxation variables is added such that Models with a greater or equal value are excluded. However, if the problem instance has a large number of relaxation variables, then adding a new constraint over these variables can lead to the exploration of a much larger search space. This article proposes new algorithms to improve the performance of linear search algorithms for MaxSAT by using Models found by the SAT solver to partition the relaxation variables. These algorithms add a new constraint on a subset of relaxation variables, thus intensifying the search on that subspace. In addition, the proposed algorithms are further enhanced by using incremental approaches where learned clauses are kept between calls to the SAT solver. Experimental results show that Model-based algorithms can outperform a classic linear search algorithm in several problem instances. Moreover, incremental approaches further improve the performance of linear search and Model-based algorithms for MaxSAT. Overall, Model-based algorithms are competitive with state-of-the-art MaxSAT solvers, and can improve their performance when solving problem instances with a large number of soft clauses.

Modern CDCL SAT solvers routinely solve very large industrial SAT instances in relatively short periods of time. It is clear that these solvers somehow exploit the structure of real-world instances. However, to-date there have been few results that precisely characterise this structure. In this paper, we provide evidence that the community structure of real-world SAT instances is correlated with the running time of CDCL SAT solvers. It has been known for some time that real-world SAT instances, viewed as graphs, have natural communities in them. A community is a sub-graph of the graph of a SAT instance, such that this sub-graph has more internal edges than outgoing to the rest of the graph. The community structure of a graph is often characterised by a quality metric called Q. Intuitively, a graph with high-quality community structure (high Q) is easily separable into smaller communities, while the one with low Q is not. We provide three results based on empirical data which show that community structure of real-world industrial instances is a better predictor of the running time of CDCL solvers than other commonly considered factors such as variables and clauses. First, we show that there is a strong correlation between the Q value and Literal Block Distance metric of quality of conflict clauses used in clause-deletion policies in Glucose-like solvers. Second, using regression analysis, we show that the the number of communities and the Q value of the graph of real-world SAT instances is more predictive of the running time of CDCL solvers than traditional metrics like number of variables or clauses. Finally, we show that randomly-generated SAT instances with 0.05 ≤ Q ≤ 0.13 are dramatically harder to solve for CDCL solvers than otherwise.

Modern SAT solvers have proved highly successful in finding counterexamples to temporal properties of systems, using a method known as model It is natural to ask whether these solvers can also be exploited for proving correctness. In fact, techniques do exist for proving properties using SAT solvers, but for the most part existing methods are either incomplete or have a low capacity relative to bounded model checking. In this paper we consider two new methods that exploit a SAT solver's ability to generate refutations in order to prove properties in an unbounded sense.

An SMT-solving procedure can be implemented by using a SAT solver to find a satisfying assignment of the propositional skeleton of the predicate formula and then deciding the feasibility of the assignment using a particular decision procedure. The complexity of the decision procedure depends on the size of the assignment. In case that the runtime of the solving is dominated by the decision procedure it is convenient to find short satisfying assignments in the SAT solving phase. Unfortunately most of the modern state-of-the-art SAT solvers always output a complete assignment of variables for satisfiable formulas even if they can be satisfied by assigning truth values to only a fraction of the variables. In this paper,we first describe an application in the code performance modeling domain, which requires SMT-solving with a costly decision procedure. Then we focus on the problem of finding minimum-size satisfying partial truth assignments. We describe and experimentally evaluate several methods how to solve this problem. These include reduction to partial maximum satisfiability - PMaxSAT, PMinSAT, pseudo-Boolean optimization and iterated SAT solving. We examine the methods experimentally on existing benchmark formulas as well as on a new benchmark set based on the performance modeling scenario.

In this paper, we present an adaptive algorithm for maximizing a monotone nonsubmodular function with a cardinality constraint. Based on the relationship between OPT and the maximum marginal gain of the elements in the ground set, the algorithm first calculates all possible values of OPT, then computes in parallel a family of sets each of which corresponding to each value of OPT, and lastly selects the set with maximum value as the desired solution. For the first, we divide the value range of OPT into finite parts and take the lower bound of each part as a possible value of OPT. As the main ingredient for the parallel computation, we use the Bernoulli distribution to independently sample elements so as to ensure further parallelism. Provided the generic submodularity ratio \(\gamma \) of the monotone set function, we prove the algorithm deserves an approximation ratio \(1-e^{-\gamma ^2}-\varepsilon \), consumes \(O(log(n/\eta )/\varepsilon ^2)\) adaptive rounds, and needs \(O(nlog(k)/\varepsilon ^3)\) oracle queries in expectation. Moreover, if the set function is submodular (i. e. \(\gamma =1\)), our algorithm can achieve an approximation guarantee \(1-1/e-\varepsilon \) coinciding with the state-of-art result.