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On optimal heuristic randomized semidecision procedures, with applications to proof complexity and cryptography
, 2010
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Optimal acceptors and optimal proof systems
"... Abstract. Unless we resolve the P vs NP question, we are unable to say whether there is an algorithm (acceptor) that accepts Boolean tautologies in polynomial time and does not accept nontautologies (with no time restriction). Unless we resolve the coNP vs NP question, we are unable to say whether ..."
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Abstract. Unless we resolve the P vs NP question, we are unable to say whether there is an algorithm (acceptor) that accepts Boolean tautologies in polynomial time and does not accept nontautologies (with no time restriction). Unless we resolve the coNP vs NP question, we are unable to say whether there is a proof system that has a polynomialsize proof for every tautology. In such a situation, it is typical for complexity theorists to search for “universal ” objects; here, it could be the “fastest ” acceptor (called optimal acceptor) and a proof system that has the “shortest ” proof (called optimal proof system) for every tautology. Neither of these objects is known to the date. In this survey we review the connections between these questions and generalizations of acceptors and proof systems that lead or may lead to universal objects. 1 Introduction and basic definitions
Time Hierarchies for Sampling Distributions
, 2012
"... We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ǫ, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1−1/k−ǫ in time t. Our proof involves reducing ..."
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We prove that for every constant k ≥ 2, every polynomial time bound t, and every polynomially small ǫ, there exists a family of distributions on k elements that can be sampled exactly in polynomial time but cannot be sampled within statistical distance 1−1/k−ǫ in time t. Our proof involves reducing the problem to a communication problem over a certain type of noisy channel. We solve the latter problem by giving a construction of a new type of listdecodable code, for a setting where there is no bound on the number of errors but each error gives more information than an erasure. 1
The Computational Complexity of Randomness
, 2013
"... This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to wh ..."
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This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized
On fast heuristic nondeterministic algorithms and short heuristic proofs
, 2013
"... In this paper we study heuristic proof systems and heuristic nondeterministic algorithms. We give an example of a language Y and a polynomialtime samplable distribution D such that the distributional problem (Y,D) belongs to the complexity class HeurNP but Y / ∈ NP if NP 6 = coNP, and (Y,D) / ∈ H ..."
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In this paper we study heuristic proof systems and heuristic nondeterministic algorithms. We give an example of a language Y and a polynomialtime samplable distribution D such that the distributional problem (Y,D) belongs to the complexity class HeurNP but Y / ∈ NP if NP 6 = coNP, and (Y,D) / ∈ HeurBPP if (NP,PSamp) 6 ⊆ HeurBPP. For a language L and a polynomial q we define the language padq(L) composed of pairs (x, r) where x is an element of L and r is an arbitrary binary string of length at least q(x). If D = {Dn}∞n=1 is an ensemble of distributions on strings, let D × U q be a distribution on pairs (x, r), where x is distributed according to Dn and r is uniformly distributed on strings of length q(n). We show that for every language L in AM there is a polynomial q such that for every distribution D concentrated on the complement of L the distributional problem (padq(L), D×U q) has a polynomially bounded heuristic proof system. Since graph nonisomorphism (GNI) is in AM, the above result is applicable to GNI. 1