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18
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.
Dual weak pigeonhole principle, pseudosurjective functions, and provability of circuit lower bounds
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Proof complexity
"... This note, based on my 4ecm lecture, exposes few basic points of proof complexity in a way accessible to any mathematician. ..."
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Cited by 12 (1 self)
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This note, based on my 4ecm lecture, exposes few basic points of proof complexity in a way accessible to any mathematician.
On optimal heuristic randomized semidecision procedures, with applications to proof complexity and cryptography
, 2010
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Constantdepth frege systems with counting axioms polynomially simulate nullstellensatz refutations. August 05 2003
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, 1998
"... Abstract. We show that constantdepth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previous ..."
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Abstract. We show that constantdepth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previously studied propositional formulas reduce to their polynomial translations. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new upper bounds on refutation sizes for certain CNFs in constantdepth Frege with counting axioms systems. 1
On the proof complexity of the NisanWigderson generator based on a hard NP ∩ coNP function
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The complexity of proving that a graph is Ramsey
, 2013
"... We say that a graph with n vertices is cRamsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is cRamsey, for every gr ..."
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Cited by 1 (1 self)
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We say that a graph with n vertices is cRamsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is cRamsey, for every graph G. Our proof makes use of the fact that every Ramsey graph must contain a large subgraph with some of the statistical properties of the random graph.
On extracting computations from propositional proofs (a survey)
, 2010
"... This paper describes a project that aims at showing that propositional proofs of certain tautologies in weak proof system give upper bounds on the computational complexity of functions associated with the tautologies. Such bounds can potentially be used to prove (conditional or unconditional) lower ..."
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This paper describes a project that aims at showing that propositional proofs of certain tautologies in weak proof system give upper bounds on the computational complexity of functions associated with the tautologies. Such bounds can potentially be used to prove (conditional or unconditional) lower bounds on the lengths of proofs of these tautologies and show separations of some weak proof systems. The prototype are the results showing the feasible interpolation property for resolution. In order to prove similar results for systems stronger than resolution one needs to define suitable generalizations of boolean circuits. We will survey the known results concerning this project and sketch in which direction we want to generalize them. 1