Results 1  10
of
54
The Complexity Of Propositional Proofs
 Bulletin of Symbolic Logic
, 1995
"... This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on ..."
Abstract

Cited by 105 (2 self)
 Add to MetaCart
This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on
Simplified and Improved Resolution Lower Bounds
 IN PROCEEDINGS OF THE 37TH IEEE FOCS
, 1996
"... We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probabili ..."
Abstract

Cited by 102 (7 self)
 Add to MetaCart
We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3CNF formula with at most n 6=5\Gammaffl clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between Resolution proof size and maximum clause size.
Integrating Equivalency Reasoning into DavisPutnam Procedure
, 2000
"... Equivalency clauses (Xors or modulo 2 arithmetics) represent a common structure in the SATencoding of many hard realworld problems and constitute a major obstacle to DavisPutnam (DP) procedure. We propose a special lookahead technique called equivalency reasoning to overcome the obstacle and ..."
Abstract

Cited by 80 (3 self)
 Add to MetaCart
Equivalency clauses (Xors or modulo 2 arithmetics) represent a common structure in the SATencoding of many hard realworld problems and constitute a major obstacle to DavisPutnam (DP) procedure. We propose a special lookahead technique called equivalency reasoning to overcome the obstacle and report on the performance of an equivalency reasoning enhanced DP procedure on SAT instances containing equivalency clauses derived from problems in parity learning, cryptographic key search and model checking. Our results show that integrating equivalency reasoning renders easy many problems which were beyond DP's reach. We also compare equivalency reasoning with general CSP lookback techniques on equivalency clauses.
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
, 2002
"... Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occu ..."
Abstract

Cited by 72 (5 self)
 Add to MetaCart
Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.
Resolution proofs of generalized pigeonhole principles
 Theoretical Computer Science
, 1988
"... We extend results of A. Haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle. These propositional formulas express the fact that there is no oneone mapping from c · n objects to n objects when c>1. As a coro ..."
Abstract

Cited by 51 (4 self)
 Add to MetaCart
We extend results of A. Haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle. These propositional formulas express the fact that there is no oneone mapping from c · n objects to n objects when c>1. As a corollary, resolution proof systems do not psimulate constant formula depth Frege proof systems. 1.
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 ..."
Abstract

Cited by 50 (3 self)
 Add to MetaCart
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 Lower Bounds for the Weak Pigeonhole Principle
, 2001
"... We prove that any Resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of ), (for some global constant ffl ? 0). One corollary is that a certain propositional formulation of the statement NP 6ae P=poly does not have short Resolution proofs. ..."
Abstract

Cited by 47 (3 self)
 Add to MetaCart
We prove that any Resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of ), (for some global constant ffl ? 0). One corollary is that a certain propositional formulation of the statement NP 6ae P=poly does not have short Resolution proofs.
Solving Difficult Instances of Boolean Satisfiability in the Presence of Symmetry
, 2002
"... Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large siz ..."
Abstract

Cited by 44 (17 self)
 Add to MetaCart
Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P##NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39, 50, 51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.
Upper and Lower Bounds for Treelike Cutting Planes Proofs
 In 9th IEEE Symposium on Logic in Computer Science
, 1994
"... In this paper we study the complexity of Cutting Planes (CP) refutations, and treelike CP refutations. Treelike CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomialsized treelike CP proofs. Our main result s ..."
Abstract

Cited by 41 (10 self)
 Add to MetaCart
In this paper we study the complexity of Cutting Planes (CP) refutations, and treelike CP refutations. Treelike CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomialsized treelike CP proofs. Our main result shows that a family of tautologies, introduced in this paper requires exponentialsized treelike CP proofs. We obtain this result by introducing a new method which relates the size of a CP refutation to the communication complexity of a related search problem. Because these tautologies have polynomialsized Frege proofs, it follows that treelike CP cannot polynomially simulate Frege systems. 1 Introduction An important open problem is to determine whether there exists a propositional proof system that admits short (polynomial size) proofs for all tautologies, or equivalently, whether or not NP equals coNP. In order to attack Research supported by NSF NYI grant CCR92570979 y Research su...
Symmetry Breaking for Boolean Satisfiability: . . .
"... Boolean Satisfiability solvers improved dramatically over the last seven years [14, 13] and are commonly used in applications such as bounded model checking, planning, and FPGA routing. However, a number of practical SAT instances remain difficult to solve. Recent work pointed out that symmetries i ..."
Abstract

Cited by 40 (9 self)
 Add to MetaCart
Boolean Satisfiability solvers improved dramatically over the last seven years [14, 13] and are commonly used in applications such as bounded model checking, planning, and FPGA routing. However, a number of practical SAT instances remain difficult to solve. Recent work pointed out that symmetries in the search space are often to blame [1]. The framework of symmetrybreaking (SBPs) [5], together with further improvements [1], was then used to achieve empirical speedups. For symmetrybreaking to be successful in practice, its overhead must be less than the complexity reduction it brings. In this work we show how logic minimization helps to improve this tradeoff and achieve much better empirical results. We also contribute detailed new studies of SBPs and their efficiency as well as new general constructions of SBPs.