Propositional SAT Solving

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.

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.

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.

SAT solving has gained tremendous interest. On the practical side there have been considerable performance improvements, due to new highly efficient algorithms, new heuristics, and optimized data structures. There are new applications and reformulations of important classical problems, mainly in the context of formal methods, where SAT solving is also applied successfully in an industrial setting. These applications range from equivalence checking, configuration, over model checking to test case generation. SAT is becoming one of the most important core technology in all these areas. Many applications actually use Satisfiability Modulo Theory (SMT), which can be seen as an extension of SAT solving. SMT has it roots in automated theorem proving. But it heavily relies on SAT technology. We discuss some key technologies in practical SAT solving, e.g. how to write a fast solver, some aspects in lifting propositional SAT technology to richer domains, how competitions can help to improve the state-of-the-art and finally touch on applications in model checking, hardware and software verification.

Propositional Satisfiability (SAT) is a keystone in the history of computer science. SAT was the first problem shown to be NP-complete in 1971 by Stephen Cook. Having passed more than 40 years from then, SAT is now a lively research field where theory and practice have a natural intermixing. In this talk, we overview the use of SAT in practical domains, where SAT is thought in a broad sense, i.e. including SAT extensions such as Maximum Satisfiability (MaxSAT), Pseudo-Boolean Optimization (PBO) and Quantified Boolean Formulas (QBF).

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.

Modern SAT solvers have proved highly successful in finding counterexamples to temporal properties of systems, using a method known as "bounded model checking". 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.

Model checking is a formal technique for automatically verifying that a finite-state model satisfies a temporal property. In model checking, generally Binary Decision Diagrams (BDDs) are used to efficiently encode the transition relation of the finite-state model. Recently model checking algorithms based on Boolean satisfiability (SAT) procedures have been developed to complement the traditional BDD-based model checking. These algorithms can be broadly classified into three categories: (1) bounded model checking which is useful for finding failures (2) hybrid algorithms that combine SAT and BDD based methods for unbounded model checking, and (3) purely SAT-based unbounded model checking algorithms. The goal of this paper is to provide a uniform and comprehensive basis for evaluating these algorithms. The paper describes eight bounded and unbounded techniques, and analyzes the performance of these algorithms on a large and diverse set of hardware benchmarks.

Satisfiability Modulo Theories (SMT) solvers have been successfully applied to solve many problems in formal verification such as bounded model checking (BMC) for many classes of systems from integrated circuits to cyber-physical systems. Typically, BMC is performed by checking satisfiability of a possibly long, but quantifier-free formula. However, BMC problems can naturally be encoded as quantified formulas over the number of BMC steps. In this approach, we then use decision procedures supporting quantifiers to check satisfiability of these quantified formulas. This approach has previously been applied to perform BMC using a Quantified Boolean Formula (QBF) encoding for purely discrete systems, and then discharges the QBF checks using QBF solvers. In this paper, we present a new quantified encoding of BMC for rectangular hybrid automata (RHA), which requires using more general logics due to the real (dense) time and real-valued state variables modeling continuous states. We have implemented a preliminary experimental prototype of the method using the HyST model transformation tool to generate the quantified BMC (QBMC) queries for the Z3 SMT solver. We describe experimental results on several timed and hybrid automata benchmarks, such as the Fischer and Lynch-Shavit mutual exclusion algorithms. We compare our approach to quantifier-free BMC approaches, such as those in the dReach tool that uses the dReal SMT solver, and the HyComp tool built on top of nuXmv that uses the MathSAT SMT solver. Based on our promising experimental results, QBMC may in the future be an effective and scalable analysis approach for RHA and other classes of hybrid automata as further improvements are made in quantifier handling in SMT solvers such as Z3.

Current algorithms for bounded model checking (BMC) use SAT methods for checking satisfiability of Boolean formulas. These BMC methods suffer from a potential memory explosion problem. Methods based on the validity of Quantified Boolean Formulas (QBF) allow an exponentially more succinct representation of the checked formulas, but have not been widely used, because of the lack of an efficient decision procedure for QBF. We evaluate the usage of QBF in BMC, using general-purpose SAT and QBF solvers. We also present a special-purpose decision procedure for QBF used in BMC, and compare our technique with the methods using general-purpose SAT and QBF solvers on real-life industrial benchmarks. Our procedure performs much better for BMC than the general-purpose QBF solvers, without incurring the space overhead of propositional SAT.

The dramatic improvements in Boolean satisfiability (SAT) solving since the turn of the millennium have made it possible to leverage state-of-the-art conflict-driven clause learning (CDCL) solvers for many combinatorial problems in academia and industry, and the use of proof logging has played a crucial role in increasing the confidence that the results these solvers produce are correct. However, the fact that SAT proof logging is performed in conjunctive normal form (CNF) clausal format means that it has not been possible to extend guarantees of correctness to the use of SAT solvers for more expressive combinatorial paradigms, where the first step is an unverified translation of the input to CNF. In this work, we show how cutting-planes-based reasoning can provide proof logging for solvers that translate pseudo-Boolean (a.k.a. 0-1 integer linear) decision problems to CNF and then run CDCL. To support a wide range of encodings, we provide a uniform and easily extensible framework for proof logging of CNF translations. We are hopeful that this is just a first step towards providing a unified proof logging approach that will also extend to maximum satisfiability (MaxSAT) solving and pseudo-Boolean optimization in general.

There has been tremendous progress in the past decade in the field of quantified Boolean formulas (QBF), both in practical solving as well as in creating a theory of corresponding proof systems and their proof complexity analysis. Both for solving and for proof complexity, it is important to have interesting formula families on which we can test solvers and gauge the strength of the proof systems. There are currently few such formula families in the literature. We initiate a general programme how to transform computationally hard problems (located in the polynomial hierarchy) into QBFs hard for the main QBF resolution systems Q-Res and QU-Res that relate to core QBF solvers. We illustrate this general approach on three problems from graph theory and logic. This yields QBF families that are provably hard for Q-Res and QU-Res (without any complexity assumptions).

A method of automatic abstraction is presented that uses proofs of unsatisfiability derived from SAT-based bounded model checking as a guide to choosing an abstraction for unbounded model checking. Unlike earlier methods, this approach is not based on analysis of abstract counterexamples. The performance of this approach on benchmarks derived from microprocessor verification indicates that SAT solvers are quite effective in eliminating logic that is not relevant to a given property. Moreover, benchmark results suggest that when bounded model checking successfully terminates, and the problem is unsatisfiable, the number of state variables in the proof of unsatisfiability tends to be small. In almost all cases tested, when bounded model checking succeeded, unbounded model checking of the resulting abstraction also succeeded.

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.

Many combinatorial problems in various fields can be translated to Maximum Satisfiability (MaxSAT) problems. Although the general problem is $\mathcal {N}\mathcal {P}$ -hard, more and more practical problems may be solved due to the significant effort which has been devoted to the development of efficient solvers. The art of constraints encoding is as important as the art of devising algorithms for MaxSAT. In this paper, we present several encoding methods of pseudo-Boolean constraints into Boolean satisfiability problems in Conjunctive Normal Form (CNF) formula, which are based on the idea of modular arithmetic and only generate auxiliary variables for each unique combination of weights. These techniques are efficient in encoding and solving MaxSAT problems. In particular, our solvers won the partial MaxSAT industrial category from 2010 through 2012 and ranked second in the 2017 main weighted track of the MaxSAT evaluation. We prove the correctness and the pseudo-polynomial space complexity of our encodings and also give a heuristics of the base selection for modular arithmetic. Our experimental results show that our encoding compactly encodes the constraints, and the obtained clauses are efficiently handled by a state-of-the-art SAT solver.