Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Gutfreund, Dan; Kawachi, Akinori

doi:10.1007/s00037-011-0010-8

Cited by 11 publications

(4 citation statements)

References 38 publications

(64 reference statements)

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“…Derandomizations of prAM are also known to imply circuit lower bounds, which are stronger than what the aforementioned results give from derandomizations of prBPP in that they either yield exponentialsize bounds [38] or give lower bounds for nondeterministic circuits [39].…”

Section: Derandomization Vs Lower Boundsmentioning

confidence: 97%

Chapter 8 Notes and References

2011

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

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Section: Derandomization Vs Lower Boundsmentioning

confidence: 97%

Chapter 8 Notes and References

2011

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

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“…Typically, the constructed pseudorandom generator is not strong enough to recover the original derandomization assumption (e.g. [IKW02,KI04,AGHK11,KvMS12,Wil13]) but some results are known that establish exact equivalence between certain sorts of derandomizations and certain sorts of pseudorandom generators (see [AvM12]). Goldreich has followed another approach [Gol11a,Gol11b] to construct pseudorandom generators from derandomization assumptions in the BPP setting.…”

Section: Derandomization Vs Pseudorandom Generatorsmentioning

confidence: 99%

Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace

Hoza¹,

Umans²

2021

SIAM J. Comput.

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Assume that for every derandomization result for logspace algorithms, there is a pseudorandom generator strong enough to nearly recover the derandomization by iterating over all seeds and taking a majority vote. We prove under a precise version of this assumption that BPL ⊆ α>0 DSPACE(log 1+α n). We strengthen the theorem to an equivalence by considering two generalizations of the concept of a pseudorandom generator against logspace. A targeted pseudorandom generator against logspace takes as input a short uniform random seed and a finite automaton; it outputs a long bitstring that looks random to that particular automaton. A simulation advice generator for logspace stretches a small uniform random seed into a long advice string; the requirement is that there is some logspace algorithm that, given a finite automaton and this advice string, simulates the automaton reading a long uniform random input. We prove that α>0 promise-BPSPACE(log 1+α n) = α>0 promise-DSPACE(log 1+α n) if and only if for every targeted pseudorandom generator against logspace, there is a simulation advice generator for logspace with similar parameters. Finally, we observe that in a certain uniform setting (namely, if we only worry about sequences of automata that can be generated in logspace), targeted pseudorandom generators against logspace can be transformed into simulation advice generators with similar parameters.

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“…For example, it is shown in (Aaronson et al 2010;Aydınlıoǧlu et al 2011) that E prSBP contains sets of circuit complexity Ω(2 n /n).…”

Section: Definitionsmentioning

confidence: 99%

The complexity of estimating min-entropy

Watson

2014

comput. complex.

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Goldreich et al. (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZK-complete. We consider the analogous problem for estimating the min-entropy and prove that it is SBP-complete, where SBP is the class of promise problems that correspond to approximate counting of NP witnesses. The result holds even when the sampling circuits are restricted to be 3-local. For logarithmic-space samplers, we observe that this problem is NP-complete by a result of Lyngsø and Pedersen on hidden Markov models (JCSS 65 (3): 2002).

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Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds

Cited by 11 publications

References 38 publications

Chapter 8 Notes and References

Chapter 8 Notes and References

Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace

The complexity of estimating min-entropy

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