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20
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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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.
Accelerating MUS extraction with recursive model rotation
 In: Proc. FMCAD’11, FMCAD (2011) 37–40
"... a wide range of practical applications. A large number of MUS extraction algorithms have been proposed over the last decade, and most of these algorithms are based on iterative calls to a SAT solver. In this paper we introduce a powerful technique for acceleration of MUS extraction algorithms called ..."
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a wide range of practical applications. A large number of MUS extraction algorithms have been proposed over the last decade, and most of these algorithms are based on iterative calls to a SAT solver. In this paper we introduce a powerful technique for acceleration of MUS extraction algorithms called recursive model rotation — a recursive version of the recently proposed model rotation technique. We demonstrate empirically that recursive model rotation leads to multiple orders of magnitude performance improvements on practical instances, and pushes the performance of our MUS extractor MUSer2 ahead of the currently available MUS extraction tools. I.
Minimal Sets over Monotone Predicates in Boolean Formulae
"... Abstract. The importance and impact of the Boolean satisfiability (SAT) problem in many practical settings is wellknown. Besides SAT, a number of computational problems related with Boolean formulas find a wide range of practical applications. Concrete examples for CNF formulas include computing pr ..."
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Abstract. The importance and impact of the Boolean satisfiability (SAT) problem in many practical settings is wellknown. Besides SAT, a number of computational problems related with Boolean formulas find a wide range of practical applications. Concrete examples for CNF formulas include computing prime implicates (PIs), minimal models (MMs), minimal unsatisfiable subsets (MUSes), minimal equivalent subsets (MESes) and minimal correction subsets (MCSes), among several others. This paper builds on earlier work by Bradley and Manna and shows that all these computational problems can be viewed as computing a minimal set subject to a monotone predicate, i.e. the MSMP problem. Thus, if cast as instances of the MSMP problem, these computational problems can be solved with the same algorithms. More importantly, the insights provided by this result allow developing a new algorithm for the general MSMP problem, that is asymptotically optimal. Moreover, in contrast with other asymptotically optimal algorithms, the new algorithm performs competitively in practice. The paper carries out a comprehensive experimental evaluation of the new algorithm on the MUS problem, and demonstrates that it outperforms state of the art MUS extraction algorithms. 1
Improving Glucose for Incremental SAT Solving with Assumptions: Application to MUS Extraction
"... Abstract. Beside the important progresses observed in SAT solving, a number of applications explicitly rely on incremental SAT solving only. In this paper, we focus on refining the incremental SAT Solver Glucose, from the SAT engine perspective, and address a number of unseen problems this new use ..."
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Abstract. Beside the important progresses observed in SAT solving, a number of applications explicitly rely on incremental SAT solving only. In this paper, we focus on refining the incremental SAT Solver Glucose, from the SAT engine perspective, and address a number of unseen problems this new use of SAT solvers opened. By playing on clause database cleaning, assumptions managements and other classical parameters, we show that our approach immediately and significantly improves an intensive assumptionbased incremental SAT solving task: Minimal Unsatisfiable Set. We believe this work could bring immediate benefits in a number of other applications relying on incremental SAT. 1
Computing small unsatisfiable cores in satisfiability modulo theories
 Journal of Artificial Intelligence Research
, 2011
"... Abstract The problem of finding small unsatisfiable cores for SAT formulas has recently received a lot of interest, mostly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be m ..."
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Abstract The problem of finding small unsatisfiable cores for SAT formulas has recently received a lot of interest, mostly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Surprisingly, the problem of finding unsatisfiable cores in SMT has received very little attention in the literature. In this paper we present a novel approach to this problem, called the LemmaLifting approach. 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, dynamically lifting the suitable amount of theory information to the Boolean level. 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 are removed. The approach is conceptually interesting, and has several advantages in practice. In fact, it is extremely simple to implement and to update, and it can be interfaced with every propositional core extractor in a plugandplay manner, so as to benefit for free of all unsatcore reduction techniques which have been or will be made available. We have evaluated our algorithm with a very extensive empirical test on SMTLIB benchmarks, which confirms the validity and potential of this approach. Motivations and Goals In the last decade we have witnessed an impressive advance in the efficiency of SAT techniques, which has brought large and previouslyintractable problems at the reach of stateoftheart SAT solvers. As a consequence, SAT solvers are now a fundamental tool in many industrialstrength applications, including most formal verification design flows for hardware systems, for equivalence, property checking, and ATPG. In particular, one of the most relevant problems in this context, thanks to its many important applications, is that of finding small unsatisfiable cores, that is, small unsatisfiable subsets of unsatisfiable sets of clauses. Surprisingly, the problem of finding unsatisfiable cores in SMT has received virtually no attention in the literature. Although some SMT tools do compute unsat cores, this is done either as a byproduct of the more general task of producing proofs, or by modifying the embedded DPLL solver so that to apply basic propositional techniques to produce an unsat core. In particular, we are not aware of any work aiming at producing small unsatisfiable cores in SMT. In this paper we present a novel approach addressing this problem, which we call the LemmaLifting approach. The main idea is to combine an SMT solver with an external propositional core extractor. The SMT solver stores and returns the theory lemmas it had to prove in order to refute the input formula; the external core extractor is then called on the Boolean abstraction of the original SMT problem and of the theory lemmas. Our algorithm is based on the following two key observations: i) the theory lemmas discovered by the SMT solver during search are valid clauses in the theory T under consideration, and therefore they do not affect the satisfiability of a formula in T ; and ii) the conjunction of the original SMT formula with all the theory lemmas is propositionally unsatisfiable. Therefore, the external (Boolean) core extractor finds an unsatisfiable core for (the Boolean abstraction of) the conjunction of the original formula and the theory lemmas, which can then be refined back into a subset of the original clauses by simply removing from it (the Boolean abstractions of) all theory lemmas. The result is an unsatisfiable core of the original SMT problem. 702 Computing Small Unsatisfiable Cores in Satisfiability Modulo Theories Although simple in principle, the approach is conceptually interesting: basically, the SMT solver is used to dynamically lift the suitable amount of theory information to the Boolean level. Furthermore, the approach has several advantages in practice: first, it is extremely simple to implement and to update; second, it is effective in finding small cores; third, the core extraction is not prone to complex SMT reasoning; finally, it can be interfaced with every propositional core extractor in a plugandplay manner, so as to benefit for free of all unsatcore reduction techniques which have been or will be made available. We have evaluated our approach by a very extensive empirical test on SMTLIB benchmarks, in terms of both effectiveness (reduction in size of the cores) and efficiency (execution time). The results confirm the validity and versatility of this approach. As a byproduct, we have also produced an extensive and insightful evaluation of the main Boolean unsatcoregeneration tools currently available. Content. The paper is organized as follows. In §2 and §3 we provide some background knowledge on techniques for SAT and SMT ( §2), and for the extraction of unsatisfiable cores in SAT and in SMT ( §3). In §4 we present and discuss our new approach and algorithm. In §5 we present and comment on the empirical tests. In §6 we conclude, suggesting some future developments.
On efficient computation of variable MUSes
 In: Proc. SAT. LNCS
, 2012
"... Abstract. In this paper we address the following problem: given an unsatisfiable CNF formula F, find a minimal subset of variables of F that constitutes the set of variables in some unsatisfiable core of F. This problem, known as variable MUS (VMUS) computation problem, captures the need to reduce ..."
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Abstract. In this paper we address the following problem: given an unsatisfiable CNF formula F, find a minimal subset of variables of F that constitutes the set of variables in some unsatisfiable core of F. This problem, known as variable MUS (VMUS) computation problem, captures the need to reduce the number of variables that appear in unsatisfiable cores. Previous work on computation of VMUSes proposed a number of algorithms for solving the problem. However, the proposed algorithms lack all of the important optimization techniques that have been recently developed in the context of (clausal) MUS computation. We show that these optimization techniques can be adopted for VMUS computation problem and result in multiple orders magnitude speedups on industrial application benchmarks. In addition, we demonstrate that in practice VMUSes can often be computed faster than MUSes, even when stateoftheart optimizations are used in both contexts. 1
Generalizing redundancy in propositional logic: Foundations and hitting sets duality
 CoRR
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Computing interpolants without proofs
 Hardware and Software: Verification and Testing, volume 7857 of Lecture Notes in Computer Science
, 2013
"... Abstract. We describe an incremental algorithm for computing interpolants for a pair ϕA, ϕB of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕA ∧ ϕB, and can be realized using any SAT solver as a black box. We ach ..."
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Abstract. We describe an incremental algorithm for computing interpolants for a pair ϕA, ϕB of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕA ∧ ϕB, and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube. 1
Formula Preprocessing in MUS Extraction
, 2013
"... Efficient algorithms for extracting minimally unsatisfiable subformulas (MUSes) of Boolean formulas find a wide range of applications in the analysis of systems, e.g., hardware and software bounded model checking. In this paper we study the applicability of preprocessing techniques for Boolean sati ..."
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Efficient algorithms for extracting minimally unsatisfiable subformulas (MUSes) of Boolean formulas find a wide range of applications in the analysis of systems, e.g., hardware and software bounded model checking. In this paper we study the applicability of preprocessing techniques for Boolean satisfiability (SAT) in the context of MUS extraction. Preprocessing has proven to be extremely important in enabling more efficient SAT solving. Hence the study of the applicability and the effectiveness of preprocessing in MUS extraction is highly relevant. Considering the extraction of both standard and group MUSes, we focus on a number of SAT preprocessing techniques, and formally prove to what extent the techniques can be directly applied in the context of MUS extraction. Furthermore, we develop a generic theoretical framework that captures MUS extraction problems, and enables formalizing conditions for correctnesspreserving applications of preprocessing techniques that are not applicable directly. We experimentally evaluate the effect of preprocessing in the context of group MUS extraction.
MUSer2: An efficient MUS extractor, system description
 JSAT
"... Algorithms for extraction of Minimally Unsatisfiable Subformulas (MUSes) of CNF formulas find a wide range of practical applications, including product configuration, knowledgebased validation, hardware and software design and verification. This paper describes the MUS extractor MUSer2. MUSer2 imple ..."
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Algorithms for extraction of Minimally Unsatisfiable Subformulas (MUSes) of CNF formulas find a wide range of practical applications, including product configuration, knowledgebased validation, hardware and software design and verification. This paper describes the MUS extractor MUSer2. MUSer2 implements a wide range of MUS extraction algorithms, integrates a number of key optimization techniques, and represents the current stateoftheart in MUS extraction. Keywords: Minimal unsatisfiability, MUS extraction, SAT applications