On the weak pigeonhole principle

Krajíček, Jan

doi:10.4064/fm170-1-8

Cited by 142 publications

(119 citation statements)

References 18 publications

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“…Finally apply CUT between this an the first hypothesis in (20), and CUT between the result and the second hypothesis in (20). This completes the proof.…”

Section: Induction Lemma 1 the Following Assertion Has Polynomial-simentioning

confidence: 73%

Mean-Payoff Games and Propositional Proofs

Atserias

Maneva

2010

Automata, Languages and Programming

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We associate a CNF-formula to every instance of the mean-payoff game problem in such a way that if the value of the game is non-negative the formula is satisfiable, and if the value of the game is negative the formula has a polynomial-size refutation in Σ 2 -Frege (i.e. DNF-resolution). This reduces mean-payoff games to the weak automatizability of Σ 2 -Frege, and to the interpolation problem for Σ 2,2 -Frege. Since the interpolation problem for Σ 1 -Frege (i.e. resolution) is solvable in polynomial time, our result is close to optimal up to the computational complexity of solving mean-payoff games. The proof of the main result requires building low-depth formulas that compute the bits of the sum of a constant number of integers in binary notation, and lowcomplexity proofs of the required arithmetic properties.

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“…Finally apply CUT between this an the first hypothesis in (20), and CUT between the result and the second hypothesis in (20). This completes the proof.…”

Section: Induction Lemma 1 the Following Assertion Has Polynomial-simentioning

confidence: 73%

Mean-Payoff Games and Propositional Proofs

Atserias

Maneva

2010

Automata, Languages and Programming

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“…Similarly to the classical case, the notion of an optimal heuristic acceptor makes sense only for languages and samplers that have no polynomially bounded heuristic acceptors, i.e., acceptors working in time polynomial in the size of the input and the parameter d. It turns out that such polynomialtime samplers and languages in co -NP roughly correspond to pseudo-random generators and the complements of their images, respectively (recall the suggestion of [ABSRW00, Kra01a,Kra01b] to consider such problems for proving lower bounds for classical proof systems). More precisely, such an intractable pair exists if and only if there is an infinitelyoften one-way function.…”

Section: Introductionmentioning

confidence: 99%

On Optimal Heuristic Randomized Semidecision Procedures, with Applications to Proof Complexity and Cryptography

et al. 2011

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The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajíček and Pudlák [KP89] show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe [Mon09] recently presented a conjecture implying that such an algorithm does not exist.We show that if one allows errors, then such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow an algorithm, called a heuristic acceptor, to claim a small number of false "theorems" and err with bounded probability on other inputs. The amount of false "theorems" is measured according to a polynomial-time samplable distribution on nontautologies. Our result remains valid for all recursively enumerable languages and can also be viewed as the existence of an optimal weakly automatizable heuristic proof system. The notion of a heuristic acceptor extends the notion of a classical acceptor; in particular, an optimal heuristic acceptor for any distribution simulates every classical acceptor for the same language.We also note that the existence of a co -NP-language L with a polynomialtime samplable distribution on L that has no polynomial-time heuristic acceptors is equivalent to the existence of an infinitely-often one-way function.

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“…The theory of proof complexity generators was developed by Krajíček [4], [19]- [21] and Alekhnovich, Ben-Sasson, Razborov, and Wigderson [22], [23]. It aims at proving lower bounds to the proof size of strong proof systems like Frege systems and their extensions which constitutes a major challenge in propositional proof complexity.…”

Section: Introductionmentioning

confidence: 99%

On the Existence of Complete Disjoint NP-Pairs

Beyersdorff

2009

2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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Abstract-Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a complete disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle).In addition, we exhibit candidates for complete NP-pairs and apply our results to a recent line of research on the construction of hard tautologies from pseudorandom generators.

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On the weak pigeonhole principle

Cited by 142 publications

References 18 publications

Mean-Payoff Games and Propositional Proofs

Mean-Payoff Games and Propositional Proofs

On Optimal Heuristic Randomized Semidecision Procedures, with Applications to Proof Complexity and Cryptography

On the Existence of Complete Disjoint NP-Pairs

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