Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, cross-fertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a black-box approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired

Abstract. Much attention has been given in recent years to the problem of

Certifying the correctness of a SAT solver is straightforward for satisfiable instances of SAT.

After establishing the unsatisfiability of a SAT instance encoding a typical design task, there is a practical need to identify its minimal unsatisfiable subsets, which pinpoint the reasons for the infeasibility of the design. Due to the potentially expensive computation, existing tools for the extraction of unsatisfiable subformulas do not guarantee the minimality of the results. This paper describes a practical algorithm that decides the minimal unsatisfiability of any CNF formula through BDD manipulation. This algorithm has a worse-case complexity that is exponential only in the treewidth of the CNF formula. We provide an empirical evaluation of the algorithm, highlighting its efficiency on a set of hard problems as well as its ability to work with existing subformula extraction tools to achieve optimal results.

The task of extracting an unsatisfiable core for a given Boolean formula has been finding more and more applications in recent years. The only existing approach that scales well for large real-world formulas exploits the ability of modern SAT solvers to produce resolution refutations. However, the resulting unsatisfiable cores are suboptimal. We propose a new algorithm for minimal unsatisfiable core extraction, based on a deeper exploration of resolution-refutation properties. Experimental results, confirming that the algorithm is able to find minimal unsatisfiable cores for well-known formal verification benchmarks, are provided.

Abstract. We address the problem of extracting Minimal Unsatisfiable Cores (MUCs) from constraint networks. This computationally hard problem has a practical interest in many application domains such as configuration, planning, diagnosis, etc. Indeed, identifying one or several disjoint MUCs can help circumscribe different sources of inconsistency in order to repair a system. In this paper, we propose an original approach that involves performing successive runs of a complete backtracking search, using constraint weighting, in order to surround an inconsistent part of a network, before identifying all transition constraints belonging to a MUC using a dichotomic process. We show the effectiveness of this approach, both theoretically and experimentally. 1

Explaining the causes of infeasibility of Boolean formulas has practical applications in various fields. We are generally interested in a minimum explanation of infeasibility that excludes irrelevant information. A

SAT is probably one of the most-studied constraint satisfaction problems. In this paper, a new hybrid technique based on local search is introduced in order to approximate and extract minimally unsatisfiable subformulas (in short, MUSes) of unsatisfiable SAT instances. It is based on an original counting heuristic grafted to a local search algorithm, which explores the neighborhood of the current interpretation in an original manner, making use of a critical clause concept. Intuitively, a critical clause is a falsified clause that becomes true thanks to a local search flip only when some other clauses become false at the same time. In the paper, the critical clause concept is investigated. It is shown to be the cornerstone of the efficiency of our approach, which outperforms competing ones to compute MUSes, inconsistent covers and sets of MUSes, most of the time. 1

Abstract. Minimally unsatisfiable subformulas (in short, MUSes) represent the smallest explanations for the inconsistency of SAT instances in terms of the number of involved clauses. Extracting MUSes can thus prove valuable because it circumscribes the sources of contradiction in an instance. In this paper, a new heuristic-based approach to approximate or compute MUSes is presented. It is shown that it often outperforms current competing ones. 1

Abstract. An autarky is a partial assignment to the variables of a Boolean CNF formula that satisfies every clause containing an assigned variable. For an unsatisfiable formula, an autarky provides information about those clauses that are essentially independent from the infeasibility; clauses satisfied by an autarky are not contained in any minimal unsatisfiable subset (MUS) or minimal correction subset (MCS) of clauses. This suggests a preprocessing step of detecting autarkies and trimming such independent clauses from an instance prior to running an algorithm for finding MUSes or MCSes. With little existing work on algorithms for finding autarkies or experimental evaluations thereof, there is room for further research in this area. Here, we present a novel algorithm that searches for autarkies directly using a standard satisfiability solver. We investigate the autarkies of several industrial benchmark suites, and experimental results show that our algorithm compares favorably to an existing approach for discovering autarkies. Finally, we explore the potential of trimming autarkies in MCS- or MUS-extraction flows. 1