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101
Short Proofs are Narrow  Resolution made Simple
 Journal of the ACM
, 2000
"... The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its treelike variant. The following consequences of these relations reveal width as a crucial " ..."
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Cited by 204 (14 self)
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The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its treelike variant. The following consequences of these relations reveal width as a crucial "resource" of Resolution proofs. In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology. In the other direction, the widthsize relations naturally suggest a simple dynamic programming procedure for automated theorem proving  one which simply searches for small width proofs. This relation guarantees that the running time (and thus the size of the produced proof) is at most quasipolynomial in the smallest treelike proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.
The efficiency of resolution and DavisPutnam procedures
 SIAM Journal on Computing
, 1999
"... We consider several problems related to the use of resolutionbased methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiabl ..."
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Cited by 66 (1 self)
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We consider several problems related to the use of resolutionbased methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiable formula of F finds a resolution proof of F , and the runtime of our algorithm is nontrivial as a function of the size of the shortest resolution proof of F . Next we investigate a class of backtrack search algorithms, commonly known as DavisPutnam procedures and provide the first averagecase complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL we prove that the running time of the algorithm on a randomly generated kCNF formula with n variables and m clauses is 2 Q(n(n/m) 1/(k2) ) with probability 1  o(1). Finally, we give new lower bounds on res(F), the size of the smallest resolution refutation ...
Stochastic Boolean Satisfiability
 Journal of Automated Reasoning
, 2000
"... . Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stoc ..."
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Cited by 65 (9 self)
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. Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSat, which can function for probabilistic domains as Sat does for deterministic domains. It shows the connection between SSat and well studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSat instances. These algorithms are validated on random SSat formulae generated under the fixedclause model. In spite of the large complexity gap between SSat (PSPACE) and Sat (NP), the paper suggests that much of what we've learned about Sat transfers to the probabilistic domain. 1. Introduction There has been a recent focus in artificial intelligence (AI) on solving problems exh...
Space Bounds for Resolution
, 1999
"... We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer ..."
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Cited by 65 (3 self)
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We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows to relate the space needed in a resolution proof of a formula to other well studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula, and as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n \Gamma c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general Pigeonhole Principle with n holes and more than n pigeons, need space n + 1 independently of the number of pigeons. Since a matching space upper bound of n + 1 for these formulas exist, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations. 3 1
Resolution is Not Automatizable Unless W[P] is Tractable
 IN 42ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2001
"... We show that neither Resolution nor treelike Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is fixedparameter tractable by randomized algorithms with onesided error. ..."
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Cited by 60 (2 self)
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We show that neither Resolution nor treelike Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is fixedparameter tractable by randomized algorithms with onesided error.
NearOptimal Separation of Treelike and General Resolution
 Electronic Colloquium in Computation Complexity
, 2000
"... We present the best known separation between treelike and general resolution, improving on the recent exp(n ) separation of [BEGJ98]. ..."
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Cited by 56 (3 self)
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We present the best known separation between treelike and general resolution, improving on the recent exp(n ) separation of [BEGJ98].
A Combinatorial Characterization of Resolution Width
 In 18th IEEE Conference on Computational Complexity
, 2002
"... We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result i ..."
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Cited by 51 (5 self)
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We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a wellknown open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the Dense Linear Order Principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the sizewidth relationship is tight.
Pseudorandom Generators Hard for kDNF Resolution and Polynomial Calculus Resolution
, 2003
"... A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the ..."
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Cited by 50 (4 self)
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A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing kDNFs instead of clauses.
On the Complexity of Unsatisfiability Proofs for Random kCNF Formulas
 In 30th Annual ACM Symposium on the Theory of Computing
, 1997
"... We study the complexity of proving unsatisfiability for random kCNF formulas with clause density D = m=n where m is number of clauses and n is the number of variables. We prove the first nontrivial general upper bound, giving algorithms that, in particular, for k = 3 produce refutations almost cer ..."
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Cited by 50 (1 self)
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We study the complexity of proving unsatisfiability for random kCNF formulas with clause density D = m=n where m is number of clauses and n is the number of variables. We prove the first nontrivial general upper bound, giving algorithms that, in particular, for k = 3 produce refutations almost certainly in time 2 O(n=D) . This is polynomial when m n 2 =logn. We show that our upper bounds are tight for certain natural classes of DavisPutnam algorithms. We show further that random 3CNF formulas of clause density D almost certainly have no resolution refutation of size smaller than 2 W(n=D 4+e ) , which implies the same lower bound on any DavisPutnam algorithm. We also give a much simpler argument based on a novel form of selfreduction that yields a slightly weaker 2 W(n=D 5+e ) lower bound. 1 Introduction The random kCNF model has been widely studied for several good reasons. First, it is an intrinsically natural model, analogous to the random graph model, that shed...
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 48 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,