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31
Lazy Satisfiability Modulo Theories
 JOURNAL ON SATISFIABILITY, BOOLEAN MODELING AND COMPUTATION 3 (2007) 141Â224
, 2007
"... Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a firstorder formula with respect to some decidable firstorder theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingl ..."
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Cited by 189 (50 self)
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Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a firstorder formula with respect to some decidable firstorder theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingly important due to its applications in many domains in different communities, in particular in formal verification. An amount of papers with novel and very efficient techniques for SMT has been published in the last years, and some very efficient SMT tools are now available. Typical SMT (T) problems require testing the satisfiability of formulas which are Boolean combinations of atomic propositions and atomic expressions in T, so that heavy Boolean reasoning must be efficiently combined with expressive theoryspecific reasoning. The dominating approach to SMT (T), called lazy approach, is based on the integration of a SAT solver and of a decision procedure able to handle sets of atomic constraints in T (Tsolver), handling respectively the Boolean and the theoryspecific components of reasoning. Unfortunately, neither the problem of building an efficient SMT solver, nor even that
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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Cited by 68 (9 self)
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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.
On finding all minimally unsatisfiable subformulas
 in Int’l Conf. on Theory and Applications of Satisfiability Testing
, 2005
"... Abstract. Much attention has been given in recent years to the problem of ..."
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Cited by 42 (4 self)
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Abstract. Much attention has been given in recent years to the problem of
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 32 (4 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 21 (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
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 20 (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.
On improving MUS extraction algorithms
 In Proc. of SAT 2011
, 2011
"... Abstract. Minimally Unsatisfiable Subformulas (MUS) find a wide range of practical applications, including product configuration, knowledgebased validation, and hardware and software design and verification. MUSes also find application in recent Maximum Satisfiability algorithms and in CNF formula ..."
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Cited by 16 (8 self)
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Abstract. Minimally Unsatisfiable Subformulas (MUS) find a wide range of practical applications, including product configuration, knowledgebased validation, and hardware and software design and verification. MUSes also find application in recent Maximum Satisfiability algorithms and in CNF formula redundancy removal. Besides direct applications in Propositional Logic, algorithms for MUS extraction have been applied to more expressive logics. This paper proposes two algorithms for MUS extraction. The first algorithm is optimal in its class, meaning that it requires the smallest number of calls to a SAT solver. The second algorithm extends earlier work, but implements a number of new techniques. The resulting algorithms achieve significant performance gains with respect to state of the art MUS extraction algorithms.
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 16 (2 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
Using unsatisfiable cores to debug multiple design errors
 in Proc. GLSVLSI, 2008
"... Due to the increasing complexity of today’s circuits a high degree of automation in the design process is mandatory. The detection of faults and design errors is supported quite well using simulation or formal verification. But locating the fault site is typically a time consuming manual task. Tech ..."
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Cited by 15 (3 self)
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Due to the increasing complexity of today’s circuits a high degree of automation in the design process is mandatory. The detection of faults and design errors is supported quite well using simulation or formal verification. But locating the fault site is typically a time consuming manual task. Techniques to automate debugging and diagnosis have been proposed. Approaches based on Boolean Satisfiability (SAT) have been demonstrated to be very effective. In this work debugging on the gate level is considered. Unsatisfiable cores contained in a SAT instance for debugging are used (1) to determine all suspects, and (2) to speedup the debugging process. In comparison to standard SATbased debugging, the experimental results show a significant speedup for debugging multiple faults.
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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Cited by 14 (4 self)
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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