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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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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.
Quantified maximum satisfiability: A coreguided approach
 In International Conference Theory and Applications of Satisfiability Testing
, 2013
"... Abstract. In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies the problem of solving quantified formulas subject to a cost function, and considers the problem in a quantified ..."
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Abstract. In recent years, there have been significant improvements in algorithms both for Quantified Boolean Formulas (QBF) and for Maximum Satisfiability (MaxSAT). This paper studies the problem of solving quantified formulas subject to a cost function, and considers the problem in a quantified MaxSAT setting. Two approaches are investigated. One is based on relaxing the soft clauses and performing a linear search on the cost function. The other approach, which is the main contribution of the paper, is inspired by recent work on MaxSAT, and exploits the iterative identification of unsatisfiable cores. The paper investigates the application of these approaches to the concrete problem of computing smallest minimal unsatisfiable subformulas (SMUS), a decision version of which is a wellknown problem in the second level of the polynomial hierarchy. Experimental results, obtained on representative problem instances, indicate that the coreguided approach for the SMUS problem outperforms the use of linear search over the values of the cost function. More significantly, the coreguided approach also outperforms the stateoftheart SMUS extractor Digger. 1
Smallest MUS Extraction with Minimal Hitting Set Dualization
"... Abstract. Minimal explanations of infeasibility are of great interest in many domains. In propositional logic, these are referred to as Minimal Unsatisfiable Subsets (MUSes). An unsatisfiable formula can have multiple MUSes, some of which provide more insights than others. Different criteria can be ..."
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Abstract. Minimal explanations of infeasibility are of great interest in many domains. In propositional logic, these are referred to as Minimal Unsatisfiable Subsets (MUSes). An unsatisfiable formula can have multiple MUSes, some of which provide more insights than others. Different criteria can be considered in order to identify a good minimal explanation. Among these, the size of an MUS is arguably one of the most intuitive. Moreover, computing the smallest MUS (SMUS) finds several practical applications that include validating the quality of the MUSes computed by MUS extractors and finding equivalent subformulae of smallest size, among others. This paper develops a novel algorithm for computing a smallest MUS, and we show that it outperforms all the previous alternatives pushing the state of the art in SMUS solving. Although described in the context of propositional logic, the presented technique can also be applied to other constraint systems. 1
Maximal falsifiability: Definitions, algorithms, and applications
 In: LPAR
, 2013
"... Abstract. Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem ..."
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Abstract. Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem has been characterized in terms of Maximal Satisfiable Subsets (MSSes) and Minimal Correction Subsets (MCSes), as well as Minimal Unsatisfiable Subsets (MUSes) and minimal hitting set dualization. However, and in contrast with MaxSAT, no such characterizations exist for MinSAT. This paper addresses this issue by casting the MinSAT problem in a more general framework. The paper studies Maximal Falsifiability, the problem of computing a subsetmaximal set of clauses that can be simultaneously falsified, and shows that MinSAT corresponds to the complement of a largest subsetmaximal set of simultaneously falsifiable clauses, i.e. the solution of the Maximum Falsifiability (MaxFalse) problem. Additional contributions of the paper include novel algorithms for Maximum and Maximal Falsifiability, as well as minimal hitting set dualization results for the MaxFalse problem. Moreover, the proposed algorithms are validated on practical instances. 1
Restoring CSP Satisfiability with MaxSAT ⋆
"... Abstract. The extraction of a Minimal Unsatisfiable Core (MUC) in a Constraint Satisfaction Problem (CSP) aims to identify a subset of constraints that make a CSP instance unsatisfiable. Recent work has addressed the identification of a Minimal Set of Unsatisfiable Tuples (MUST) in order to restore ..."
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Abstract. The extraction of a Minimal Unsatisfiable Core (MUC) in a Constraint Satisfaction Problem (CSP) aims to identify a subset of constraints that make a CSP instance unsatisfiable. Recent work has addressed the identification of a Minimal Set of Unsatisfiable Tuples (MUST) in order to restore the CSP satisfiability with respect to that MUC. A twostep algorithm has been proposed: first, a MUC is identified, and second, a MUST in the MUC is identified. This paper proposes an integrated algorithm for restoring satisfiability in a CSP, making use of an unsatisfiabilitybased MaxSAT solver. The proposed approach encodes the CSP instance as a partial MaxSAT instance, in such a way that solving the MaxSAT instance corresponds to identifying the smallest set of tuples to be removed from the CSP instance to restore satisfiability. Experimental results illustrate the feasibility of the approach. Key words: constraint satisfaction problems, minimal unsatisfiable cores, minimal set of unsatisfiable tuples, maximum satisfiability 1
SATBased Formula Simplification
"... Abstract. The problem of propositional formula minimization can be traced to the mid of the last century, to the seminal work of Quine and McCluskey, with a large body of work ensuing from this seminal work. Given a set of implicants (or implicates) of a formula, the goal for minimization is to find ..."
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Abstract. The problem of propositional formula minimization can be traced to the mid of the last century, to the seminal work of Quine and McCluskey, with a large body of work ensuing from this seminal work. Given a set of implicants (or implicates) of a formula, the goal for minimization is to find a smallest set of prime implicants (or implicates) equivalent to the original formula. This paper considers the more general problem of computing a smallest prime representation of a nonclausal propositional formula, which we refer to as formula simplification. Moreover, the paper proposes a novel, entirely SATbased, approach for the formula simplification problem. The original problem addressed by the QuineMcCluskey procedure can thus be viewed as a special case of the problem addressed in this paper. Experimental results, obtained on wellknown representative problem instances, demonstrate that a SATbased approach for formula simplification is a viable alternative to existing implementations of the QuineMcCluskey procedure. 1
On computing Minimal Independent Support and its applications to sampling and counting?
"... Abstract. Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions t ..."
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Abstract. Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in stateoftheart constrained sampling and counting tools, while still retaining theoretical guarantees. 1
1Maximal Falsifiability Definitions, Algorithms, and Applications
"... Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem has been ..."
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Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem has been characterized in terms of Maximal Satisfiable Subsets (MSSes) and Minimal Correction Subsets (MCSes), as well as Minimal Unsatisfiable Subsets (MUSes) and minimal hitting set dualization. However, and in contrast with MaxSAT, no such characterizations exist for MinSAT. This paper addresses this issue by casting the MinSAT problem in a more general framework. The paper studies Maximal Falsifiability, the problem of computing a subsetmaximal set of clauses that can be simultaneously falsified, and shows that MinSAT corresponds to the complement of a largest subsetmaximal set of simultaneously falsifiable clauses, i.e. the solution of the Maximum Falsifiability (MaxFalse) problem. Additional contributions of the paper include novel algorithms for Maximum and Maximal Falsifiability, as well as minimal hitting set dualization results for the MaxFalse problem. Moreover, the proposed algorithms are validated on practical instances.