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24
Towards understanding and harnessing the potential of clause learning
 Journal of Artificial Intelligence Research
, 2004
"... Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant realworld problems, such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitat ..."
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Cited by 96 (10 self)
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Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant realworld problems, such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitations of the technique. This paper presents the first precise characterization of clause learning as a proof system (CL), and begins the task of understanding its power by relating it to the wellstudied resolution proof system. In particular, we show that with a new learning scheme, CL can provide exponentially shorter proofs than many proper refinements of general resolution (RES) satisfying a natural property. These include regular and DavisPutnam resolution, which are already known to be much stronger than ordinary DPLL. We also show that a slight variant of CL with unlimited restarts is as powerful as RES itself. Translating these analytical results to practice, however, presents a challenge because of the nondeterministic nature of clause learning algorithms. We propose a novel way of exploiting the underlying problem structure, in the form of a high level problem description such as a graph or PDDL specification, to guide clause learning algorithms toward faster solutions. We show that this leads to exponential speedups on grid and randomized pebbling problems, as well as substantial improvements on certain ordering formulas. 1.
EFFICIENT ALGORITHMS FOR CLAUSELEARNING SAT SOLVERS
, 2004
"... Boolean satisfiability (SAT) is NPcomplete. No known algorithm for SAT is of polynomial time complexity. Yet, many of the SAT instances generated as a means of solving realworld electronic design automation problems are simple enough, structurally, that modern solvers can decide them efficiently. ..."
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Cited by 73 (0 self)
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Boolean satisfiability (SAT) is NPcomplete. No known algorithm for SAT is of polynomial time complexity. Yet, many of the SAT instances generated as a means of solving realworld electronic design automation problems are simple enough, structurally, that modern solvers can decide them efficiently. Consequently, SAT solvers are widely used in industry for logic verification. The most robust solver algorithms are poorly understood and only vaguely described in the literature of the field. We refine these algorithms, and present them clearly. We introduce several new techniques for Boolean constraint propagation that substantially improve solver efficiency. We explain why literal count decision strategies succeed, and on that basis, we introduce a new decision strategy that outperforms the state of the art. The culmination of this work is the most powerful SAT solver publically available.
New inference rules for MaxSAT
 JAIR
, 2007
"... Abstract. Exact MaxSAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the MaxSAT problem for the simplified for ..."
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Cited by 42 (9 self)
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Abstract. Exact MaxSAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the MaxSAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of original inference rules that, besides being applied efficiently, transform MaxSAT instances into equivalent MaxSAT instances which are easier to solve. The soundness of the rules, that can be seen as refinements of unit resolution adapted to MaxSAT, are proved in a novel and simple way via an integer programming transformation. Aiming to find out how powerful the inference rules are in practice, we have developed a new MaxSAT solver, called MaxSatz, which incorporates those rules, and performed an experimental investigation. The results obtained provide empirical evidence that MaxSatz is very competitive and greatly outperforms the best stateoftheart MaxSAT solvers on random Max2SAT, random Max3SAT, MaxCut, and Graph 3coloring instances, as well as benchmarks submitted to the MaxSAT Evaluation 2006. 1
Clauselearning algorithms with many restarts and boundedwidth resolution
, 2009
"... We offer a new understanding of some aspects of practical SATsolvers that are based on DPLL with unitclause propagation, clauselearning, and restarts. On the theoretical side, we do so by analyzing a concrete algorithm which we claim is faithful to what practical solvers do. In particular, befo ..."
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Cited by 33 (2 self)
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We offer a new understanding of some aspects of practical SATsolvers that are based on DPLL with unitclause propagation, clauselearning, and restarts. On the theoretical side, we do so by analyzing a concrete algorithm which we claim is faithful to what practical solvers do. In particular, before making any new decision or restart, the solver repeatedly applies the unitresolution rule until saturation, and leaves no component to the mercy of nondeterminism except for some internal randomness. We prove the perhaps surprising fact that, although the solver is not explicitely designed for it, it ends up behaving as widthk resolution after no more than n 2k+1 conflicts and restarts, where n is the number of variables. In other words, widthk resolution can be thought as n 2k+1 restarts of the unitresolution rule with learning. On the experimental side, we give evidence for the claim that this theoretical result describes real world solvers. We do so by running some of the most prominent solvers on some CNF formulas that we designed to have resolution refutations of width k. It turns out that the upper bound of the theoretical result holds for these solvers and that the true performance appears to be not very far from it.
RSat 2.0: SAT Solver Description
, 2007
"... RSat 2.0 is a DPLLbased complete SAT solver that employs many modern techniques such as those used in MiniSat [3] and Chaff [8]. RSat 2.0 is an improved version of RSat [10], which won the third place in the SATRace 2006 competition [11]. While RSat 2.0 is designed to perform best on industrial SA ..."
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Cited by 31 (3 self)
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RSat 2.0 is a DPLLbased complete SAT solver that employs many modern techniques such as those used in MiniSat [3] and Chaff [8]. RSat 2.0 is an improved version of RSat [10], which won the third place in the SATRace 2006 competition [11]. While RSat 2.0 is designed to perform best on industrial SAT instances,
Complete local search for propositional satisfiability
 In proceedings of AAAI
, 2004
"... Algorithms based on following local gradient information are surprisingly effective for certain classes of constraint satisfaction problems. Unfortunately, previous local search algorithms are notoriously incomplete: They are not guaranteed to find a feasible solution if one exists and they cannot b ..."
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Cited by 27 (0 self)
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Algorithms based on following local gradient information are surprisingly effective for certain classes of constraint satisfaction problems. Unfortunately, previous local search algorithms are notoriously incomplete: They are not guaranteed to find a feasible solution if one exists and they cannot be used to determine unsatisfiability. We present an algorithmic framework for complete local search and discuss in detail an instantiation for the propositional satisfiability problem (SAT). The fundamental idea is to use constraint learning in combination with a novel objective function that converges during search to a surface without local minima. Although the algorithm has worstcase exponential space complexity, we present empirical results on challenging SAT competition benchmarks that suggest that our implementation can perform as well as stateoftheart solvers based on more mature techniques. Our framework suggests a range of possible algorithms lying between treebased search and local search.
Memoization and DPLL: Formula caching proof systems
 In Proceedings 18th Annual IEEE Conference on Computational Complexity
, 2003
"... A fruitful connection between algorithm design and proof complexity is the formalization of the ¤¦¥¨§© § approach to satisfiability testing in terms of treelike resolution proofs. We consider extensions of the ¤¦¥¨§© § approach that add some version of memoization, remembering formulas the algorith ..."
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Cited by 23 (9 self)
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A fruitful connection between algorithm design and proof complexity is the formalization of the ¤¦¥¨§© § approach to satisfiability testing in terms of treelike resolution proofs. We consider extensions of the ¤¦¥¨§© § approach that add some version of memoization, remembering formulas the algorithm has previously shown unsatisfiable. Various versions of such formula caching algorithms have been suggested for satisfiability and stochastic satisfiability ([10, 1]). We formalize this method, and characterize the strength of various versions in terms of proof systems. These proof systems seem to be both new and simple, and have a rich structure. We compare their strength to several studied proof systems: treelike resolution, regular resolution, general resolution, and ���������� �. We give both simulations and separations. 1
Short proofs may be spacious: An optimal separation of space and length in resolution
, 2008
"... A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negat ..."
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Cited by 22 (10 self)
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A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n / log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the blackwhite pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and H˚astad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard blackwhite pebbling price.
Towards an Optimal Separation of Space and Length in Resolution
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2008
"... Most stateoftheart satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory corre ..."
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Cited by 14 (9 self)
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Most stateoftheart satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ ( √ n) on the space needed for socalled pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width in (Nordström 2006) from logarithmic to polynomial. Also, continuing the line of research initiated by (BenSasson 2002) into tradeoffs between different proof complexity measures, we present a simplified proof of the recent lengthspace tradeoff result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential tradeoffs in resolution.
Using Problem Structure for Efficient Clause Learning
 In Proceedings of the 6th International Conference on Theory and Applications of Satisfiability Testing
, 2003
"... DPLL based clause learning algorithms for satisfiability testing are known to work very well in practice. However, like most branchandbound techniques, their performance depends heavily on the variable order used in making branching decisions. We propose a novel way of exploiting the underlying ..."
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Cited by 14 (4 self)
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DPLL based clause learning algorithms for satisfiability testing are known to work very well in practice. However, like most branchandbound techniques, their performance depends heavily on the variable order used in making branching decisions. We propose a novel way of exploiting the underlying problem structure to guide clause learning algorithms toward faster solutions. The key idea is to use a higher level problem description, such as a graph or a PDDL specification, to generate a good branching sequence as an aid to SAT solvers.