Results 1  10
of
43
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 113 (3 self)
 Add to MetaCart
(Show Context)
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
A Fast PseudoBoolean Constraint Solver
, 2003
"... Linear PseudoBoolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linea ..."
Abstract

Cited by 110 (1 self)
 Add to MetaCart
Linear PseudoBoolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linearly combined, parameterized weights, yet also offer efficient search strategies for proving or disproving whether a satisfying solution exists. Furthermore, corresponding decision procedures can easily be extended for minimizing or maximizing an LPB objective function, thus providing a core optimization method for many problems in logic and physical synthesis. In this paper we review how recent advances in satisfiability (SAT) search can be extended for pseudoBoolean constraints and describe a new LPB solver that is based on generalized constraint propagation and conflictbased learning. We present a comparison with other, stateoftheart LPB solvers which demonstrates the overall efficiency of our method.
Polynomial size proofs of the propositional pigeonhole principle
 Journal of Symbolic Logic
, 1987
"... Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege syste ..."
Abstract

Cited by 77 (7 self)
 Add to MetaCart
(Show Context)
Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in computational complexity such as whether NP = coNP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which
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 ..."
Abstract

Cited by 67 (1 self)
 Add to MetaCart
(Show Context)
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 ...
Lower bounds on Hilbert's Nullstellensatz and propositional proofs
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 1996
"... The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polyno ..."
Abstract

Cited by 64 (19 self)
 Add to MetaCart
(Show Context)
The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polynomials P,(x) such that £, P,(x)Qt(x) = 1. This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into ^element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count^fo,...) with underlying variables xe, where e ranges over <7element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count $ n+I do not have polynomial size, constantdepth Frege proofs from substitution instances of Count/?, where m^O (modp). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Count, can be proved efficiently from Countp, for all values of p and q.
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 ..."
Abstract

Cited by 47 (0 self)
 Add to MetaCart
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,
The Asymptotic Order of the Random kSAT Threshold
 In Proc. FOCS
, 2002
"... Form a random kSAT formula on n variables by selecting uniformly and independently m = rn clauses out of all 2 possible kclauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r k such that, as n tends to infinity, the probability that the formula is sa ..."
Abstract

Cited by 44 (13 self)
 Add to MetaCart
Form a random kSAT formula on n variables by selecting uniformly and independently m = rn clauses out of all 2 possible kclauses. The Satisfiability Threshold Conjecture asserts that for each k there exists a constant r k such that, as n tends to infinity, the probability that the formula is satisfiable tends to 1 if r < r k and to 0 if r > r k . It has long been known that 2 =k < r k < 2 . We prove that r k > 2 ln 2 d k , where d k ! (1 + ln 2)=2. Our proof also allows a blurry glimpse of the "geometry" of the set of satisfying truth assignments.
Random kSAT: two moments suffice to cross a sharp threshold
 CoRR
, 2006
"... Abstract. Many NPcomplete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that abo ..."
Abstract

Cited by 42 (4 self)
 Add to MetaCart
(Show Context)
Abstract. Many NPcomplete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2colorability (Property B) and random NotAllEqual kSAT is 2 k−1 ln 2 − O(1). As a corollary, we establish that the threshold for random kSAT is of order Θ(2 k), resolving a longstanding open problem.