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16
Picosat essentials
 Journal on Satisfiability, Boolean Modeling and Computation (JSAT
"... In this article we describe and evaluate optimized compact data structures for watching literals. Experiments with our SAT solver PicoSAT show that this lowlevel optimization not only saves memory, but also turns out to speed up the SAT solver considerably. We also discuss how to store proof traces ..."
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Cited by 90 (11 self)
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In this article we describe and evaluate optimized compact data structures for watching literals. Experiments with our SAT solver PicoSAT show that this lowlevel optimization not only saves memory, but also turns out to speed up the SAT solver considerably. We also discuss how to store proof traces compactly in memory and further unique features of PicoSAT including an aggressive restart schedule. Keywords: SAT solver, watched literals, occurrence lists, proof traces, restarts
A first step towards a unified proof checker for QBF
 In Tenth International Conference on Theory and Applications of Satisfiability Testing (SAT
, 2007
"... Abstract. Compared to SAT, there is no simple concept of what a solution to a QBF problem is. Furthermore, as the series of QBF evaluations shows, the QBF solvers that are available often disagree. Thus, proof generation for QBF seems to be even more important than for SAT. In this paper we propose ..."
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Cited by 9 (5 self)
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Abstract. Compared to SAT, there is no simple concept of what a solution to a QBF problem is. Furthermore, as the series of QBF evaluations shows, the QBF solvers that are available often disagree. Thus, proof generation for QBF seems to be even more important than for SAT. In this paper we propose a new uniform proof format, which captures refutations and witnesses for a variety of QBF solvers, and is based on a novel extended resolution rule for QBF. Our experiments show the flexibility of this new format. We also identify shortcomings of our format and conjecture that a purely resolution based proof calculus is not powerful enough to trace the most efficient solvers. 1
Compressing propositional refutations
 Sixth International Workshop on Automated Verification of Critical Systems (AVOCS ’06) – Preliminary Proceedings
, 2006
"... We report initial results on shortening propositional resolution refutation proofs. This has an application in speeding up deductive reconstruction (in theorem provers) of large propositional refutations, such as those produced by SATsolvers. Key words: Proof verification, Propositional refutations ..."
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Cited by 6 (0 self)
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We report initial results on shortening propositional resolution refutation proofs. This has an application in speeding up deductive reconstruction (in theorem provers) of large propositional refutations, such as those produced by SATsolvers. Key words: Proof verification, Propositional refutations 1
Extended Clause Learning
"... The past decade has seen clause learning as the most successful algorithm for SAT instances arising from realworld applications. This practical success is accompanied by theoretical results showing clause learning as equivalent in power to resolution. There exist, however, problems that are intract ..."
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Cited by 5 (0 self)
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The past decade has seen clause learning as the most successful algorithm for SAT instances arising from realworld applications. This practical success is accompanied by theoretical results showing clause learning as equivalent in power to resolution. There exist, however, problems that are intractable for resolution, for which clauselearning solvers are hence doomed. In this paper, we present extended clause learning, a practical SAT algorithm that surpasses resolution in power. Indeed, we prove that it is equivalent in power to extended resolution, a proof system strictly more powerful than resolution. Empirical results based on an initial implementation suggest that the additional theoretical power can indeed translate into substantial practical gains. 1.
Nearlyexponential size lower bounds for symbolic quantifier elimination algorithms and OBDDbased proofs of unsatisfiability
 Electronic Colloquium on Computational Complexity
, 2007
"... We demonstrate a family of propositional formulas in conjunctive normal form so that a formula of size N requires size 2 Ω ( 7 √ N/logN) to refute using the treelike OBDD refutation system of Atserias, Kolaitis and Vardi [3] with respect to all variable orderings. All known symbolic quantifier elim ..."
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Cited by 3 (1 self)
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We demonstrate a family of propositional formulas in conjunctive normal form so that a formula of size N requires size 2 Ω ( 7 √ N/logN) to refute using the treelike OBDD refutation system of Atserias, Kolaitis and Vardi [3] with respect to all variable orderings. All known symbolic quantifier elimination algorithms for satisfiability generate treelike proofs when run on unsatisfiable CNFs, so this lower bound applies to the runtimes of these algorithms. Furthermore, the lower bound generalizes earlier results on OBDDbased proofs of unsatisfiability in that it applies for all variable orderings, it applies when the clauses are processed according to an arbitrary schedule, and it applies when variables are eliminated via quantification. 1
Enhancing Unsatisfiable Cores for LTL with Information on Temporal Relevance
, 2013
"... LTL is frequently used to express specifications in many domains such as embedded systems or business processes. Witnesses can help to understand why an LTL specification is satisfiable, and a number of approaches exist to make understanding a witness easier. In the case of unsatisfiable specificati ..."
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LTL is frequently used to express specifications in many domains such as embedded systems or business processes. Witnesses can help to understand why an LTL specification is satisfiable, and a number of approaches exist to make understanding a witness easier. In the case of unsatisfiable specifications unsatisfiable cores (UCs), i.e., parts of an unsatisfiable formula that are themselves unsatisfiable, are a well established means for debugging. However, little work has been done to help understanding a UC of an unsatisfiable LTL formula. In this paper we suggest to enhance a UC of an unsatisfiable LTL formula with additional information about the time points at which the subformulas of the UC are relevant for unsatisfiability. For example, in (Gp) ∧ (X¬p) the first occurrence of p is really only “relevant ” for unsatisfiability at time point 1 (time starts at time point 0). We present a method to extract such information from the resolution graph of a temporal resolution proof of unsatisfiability of an LTL formula. We implement our method in TRP++, and we experimentally evaluate it. Source code of our tool is available. 1
AVoCS 2006 Compressing Propositional Refutations
"... We report initial results on shortening propositional resolution refutation proofs. This has an application in speeding up deductive reconstruction (in theorem provers) of large propositional refutations, such as those produced by SATsolvers. ..."
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We report initial results on shortening propositional resolution refutation proofs. This has an application in speeding up deductive reconstruction (in theorem provers) of large propositional refutations, such as those produced by SATsolvers.
On Resolution Proofs for Combinational Equivalence ABSTRACT
"... Modern combinational equivalence checking (CEC) engines are complicated programs which are difficult to verify. In this paper we show how a modern CEC engine can be modified to produce a proof of equivalence when it proves a miter unsatisfiable. If the CEC engine formulates the problem as a single S ..."
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Modern combinational equivalence checking (CEC) engines are complicated programs which are difficult to verify. In this paper we show how a modern CEC engine can be modified to produce a proof of equivalence when it proves a miter unsatisfiable. If the CEC engine formulates the problem as a single SAT instance (call this naïve), one can use the resolution proof of unsatisfiability as a proof of equivalence. However, a modern CEC engine does not directly invoke a SAT solver for the whole miter, but instead uses a variety of techniques such as structural hashing, detection of intermediate functional equivalences, and circuit rewriting to first simplify the problem. We show that in spite of using these simplification techniques, a CEC engine can be modified to generate a single (extended) resolution proof for the whole miter just as in the naïve case. The benefit of having a single proof is that the proof verification program remains extremely simple, and its correctness is much easier to establish than that of the CEC engine.
1 Introduction Compressing Propositional Proofs by Common Subproof Extraction
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AFirstStepTowards a Unified Proof Checker for QBF
"... Abstract. Compared to SAT, there is no simple concept of what a solution to a QBF problem is. Furthermore, as the series of QBF evaluations shows, the QBF solvers that are available often disagree. Thus, proof generation for QBF seems to be even more important than for SAT. In this paper we propose ..."
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Abstract. Compared to SAT, there is no simple concept of what a solution to a QBF problem is. Furthermore, as the series of QBF evaluations shows, the QBF solvers that are available often disagree. Thus, proof generation for QBF seems to be even more important than for SAT. In this paper we propose a new uniform proof format, which captures refutations and witnesses for a variety of QBF solvers, and is based on a novel extended resolution rule for QBF. Our experiments show the flexibility of this new format. We also identify shortcomings of our format and conjecture that a purely resolution based proof calculus is not powerful enough to trace the most efficient solvers. 1