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Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints
"... Abstract. 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 ..."
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Abstract. 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 useful 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. 1.
Propositional Satisfiability and Constraint Programming: a Comparative Survey
 ACM Computing Surveys
, 2006
"... Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms ..."
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Cited by 34 (4 self)
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Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising 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 blackbox 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
A scalable algorithm for minimal unsatisfiable core extraction
 IN PROC. SAT’06
, 2006
"... 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 realworld formulas exploits the ability of modern SAT solvers to produce resolution refutations. However, the ..."
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Cited by 22 (3 self)
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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 realworld 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 resolutionrefutation properties. Experimental results, confirming that the algorithm is able to find minimal unsatisfiable cores for wellknown formal verification benchmarks, are provided.
A Simple and Flexible Way of Computing Small Unsatisfiable Cores in SAT Modulo Theories
 IN: PROCEEDINGS OF THE 10TH INTERNATIONAL CONFERENCE ON THEORY AND APPLICATIONS OF SATISFIABILITY TESTING (SAT2007
, 2007
"... Finding small unsatisfiable cores for SAT problems has recently received a lot of interest, mostly for its applications in formal verification. Surprisingly, the same problem in the context of SAT Modulo Theories (SMT) has instead received very little attention in the literature; in particular, we ..."
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Cited by 15 (3 self)
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Finding small unsatisfiable cores for SAT problems has recently received a lot of interest, mostly for its applications in formal verification. Surprisingly, the same problem in the context of SAT Modulo Theories (SMT) has instead received very little attention in the literature; in particular, we are not aware of any work aiming at producing small unsatisfiable cores in SMT. The purpose of this paper is to start filling the gap in this area, by proposing a novel approach for computing small unsat cores in SMT. The main idea is to combine an SMT solver with an external propositional core extractor: the SMT solver produces the theory lemmas found during the search; the core extractor is then called on the boolean abstraction of the original SMT problem and of the theory lemmas. This results in an unsatisfiable core for the original SMT problem, once the remaining theory lemmas have been removed. The approach has several advantages: it is extremely simple to implement
Extracting MUCs from constraint networks
 In Proceedings of ECAI’06
, 2006
"... 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 hel ..."
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Cited by 14 (5 self)
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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
LocalSearch Extraction of MUSes
"... SAT is probably one of the moststudied 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 cou ..."
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Cited by 12 (1 self)
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SAT is probably one of the moststudied 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
A branchandbound algorithm for extracting smallest minimal unsatisfiable formulas
 In International Conference on Theory and Applications of Satisfiability Testing (SAT’05
, 2005
"... Abstract. We tackle the problem of finding a smallestcardinality MUS (SMUS) of a given formula. The SMUS provides a succinct explanation of infeasibility and is valuable for applications that rely on such explanations. We present a branchandbound algorithm that utilizes iterative MAXSAT solutions ..."
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Abstract. We tackle the problem of finding a smallestcardinality MUS (SMUS) of a given formula. The SMUS provides a succinct explanation of infeasibility and is valuable for applications that rely on such explanations. We present a branchandbound algorithm that utilizes iterative MAXSAT solutions to generate lower and upper bounds on the size of the SMUS, and branch on specific subformulas to find it. We report experimental results on formulas from DIMACS and DaimlerChrysler product configuration suites. 1
Extracting Minimum Unsatisfiable Cores With a Greedy Genetic Algorithm
 IN PROC. ACAI’06
, 2006
"... 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 ..."
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Cited by 11 (3 self)
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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
Boosting minimal unsatisfiable core extraction
 in FMCAD, 2010
"... Abstract—A variety of tasks in formal verification require finding small or minimal unsatisfiable cores (subsets) of an unsatisfiable set of constraints. This paper proposes two algorithms for finding a minimal unsatisfiable core or, if a timeout occurs, a small nonminimal unsatisfiable core. Our ..."
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Cited by 10 (0 self)
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Abstract—A variety of tasks in formal verification require finding small or minimal unsatisfiable cores (subsets) of an unsatisfiable set of constraints. This paper proposes two algorithms for finding a minimal unsatisfiable core or, if a timeout occurs, a small nonminimal unsatisfiable core. Our algorithms can be applied to either standard clauselevel unsatisfiable core extraction or highlevel unsatisfiable core extraction, that is, an extraction of an unsatisfiable core in terms of “interesting” propositional constraints supplied by the user application. We demonstrate that one of our algorithms outperforms existing algorithms for clauselevel minimal unsatisfiable core extraction on large wellknown industrial benchmarks. We also show that our algorithms are highly scalable for the problem of highlevel minimal unsatisfiable core extraction on huge benchmarks generated by Intel’s proofbased abstraction refinement flow. In addition, we provide a comparative analysis of the impact of various algorithms on unsatisfiable core extraction. I.
Extracting MUSes
 Proc. of the 17 th European Conference on Artificial Intelligence (ECAI’2006
, 2006
"... 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 thi ..."
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Cited by 8 (5 self)
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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 heuristicbased approach to approximate or compute MUSes is presented. It is shown that it often outperforms current competing ones. 1