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52
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 34 (3 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.
The complexity of propositional proofs
 BULLETIN OF SYMBOLIC LOGIC
"... Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. Thi ..."
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Cited by 31 (0 self)
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Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.
On the Complexity of Resolution with Bounded Conjunctions
, 2004
"... We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Krajicek in [20] which extends Resolution by allowing disjunctions of conjunctions of up to k * 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to pr ..."
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Cited by 28 (4 self)
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We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Krajicek in [20] which extends Resolution by allowing disjunctions of conjunctions of up to k * 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreover Resolution, while simulating treelike Res(k), is almost exponentially separated from treelike Res(k). To study space complexity
Constraint Propagation as a Proof System
 10th Int.Conf. on Principles and Practice of Constraint Programing, LN in Computer Science vol.3258
, 2004
"... Refutation proofs can be viewed as a special case of constraint propagation, which is a fundamental technique in solving constraintsatisfaction problems. ..."
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Cited by 27 (1 self)
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Refutation proofs can be viewed as a special case of constraint propagation, which is a fundamental technique in solving constraintsatisfaction problems.
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 23 (11 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.
Understanding Space in Proof Complexity: Separations and Tradeoffs via Substitutions
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 125 (2010)
, 2010
"... For current stateoftheart satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunct ..."
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Cited by 21 (10 self)
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For current stateoftheart satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are tradeoffs between these two measures, has remained essentially open apart from a few results in restricted settings. In this paper, we remedy this situation by proving a host of lengthspace tradeoff results for resolution in a completely general setting. Our collection of tradeoffs cover almost the whole range of values for the space complexity of formulas, and most of the tradeoffs are superpolynomial or even exponential and essentially tight. Using similar techniques, we show that these tradeoffs in fact extend (albeit with worse parameters) to the exponentially stronger kDNF resolution proof systems, which operate with formulas in disjunctive normal form with terms of bounded arity k. We also answer the open question
Narrow proofs may be spacious: Separating space and width in resolution (Extended Abstract)
 REVISION 02, ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2005
"... The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously ..."
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Cited by 20 (7 self)
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The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of kCNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.
PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when compari ..."
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Cited by 18 (6 self)
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Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and tradeoffs between proof size and proof space.
Approximability and proof complexity
, 2012
"... This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Pa ..."
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Cited by 17 (6 self)
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This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any nvariable degreed proof can be found in time n O(d). Furthermore, the SDP is dual to the wellknown Lasserre SDP hierarchy, meaning that the “d/2round Lasserre value ” of an optimization problem is equal to the best bound provable using a degreed SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known “hard instances ” for the UniqueGames problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the BalancedSeparator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot–Vishnoi MaxCut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor.952 (>.878) using a constantdegree proof. These investigations also raise an interesting mathematical question: is there a constantdegree SOS proof of the Central Limit Theorem?