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Fefferman, Bill; Shaltiel, Ronen; Umans, Christopher; Viola, Emanuele

doi:10.4086/toc.2013.v009a026

Cited by 18 publications

(6 citation statements)

References 36 publications

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“…The class of small-depth Boolean circuits (AC 0 ) is a class of central interest in unconditional derandomization, and has been the subject of intensive research in this area over the past 30 years [6,49,57,58,54,53,45,72,74,13,62,21,46,29,2,1,69,51,31,35,34,73,33,70,36,24]. This highly successful line of work on derandomizing AC 0 has generated a wealth of ideas and techniques that have become mainstays in the field of pseudorandomness.…”

Section: Prgs For Ac 0 Circuitsmentioning

confidence: 99%

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Servedio¹,

Tan²

2022

Theory of Comput.

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We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse F 2 polynomials. Our main results are an ε-PRG for the class of size-M depth-d AC 0 circuits with seed length log(M) d+O(1) • log(1/ε), and an ε-PRG for the class of S-sparse F 2 polynomials with seed length 2 O( √ log S) • log(1/ε). These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: substantially improving on the seed lengths of either PRG would require a breakthrough on longstanding and notorious circuit lower bound problems.The key enabling ingredient in our approach is a new pseudorandom multi-switching lemma. We derandomize recently developed multi-switching lemmas, which are powerful generalizations of Håstad's switching lemma that deal with families of depth-two circuits. Our pseudorandom multi-switching lemma-a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family-achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and

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Section: Prgs For Ac 0 Circuitsmentioning

confidence: 99%

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Servedio¹,

Tan²

2022

Theory of Comput.

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“…which negates that H is an ε-hitting set for 1-branching programs of width 3 according to (4). In fact, P implements the corresponding negated conjunction of DNF and CNF (10). We assume q ≥ 1, r ≥ 1, and |Q q | > 1, while the proof for q = 0 or r = 0 or |Q q | = 1 is similar.…”

Section: Proofmentioning

confidence: 99%

“…2 . This implements DNF monomials ∧ i∈Q j ℓ(x i ) in (10) which are candidates for the monomial ∧ i∈Q ℓ(x i ) in (19).…”

Section: The Plan Of the Proofmentioning

confidence: 99%

“…Recently, considerable attention has been paid to improving the seed length to O(log n) in the constant-width case, which is a fundamental problem with many applications in circuit lower bounds and derandomization [5,6]. The problem has been resolved for width 2 but the known techniques provably fail for width 3 [7,8,9,10,5,6], which applies even to hitting set generators [8].…”

Section: Introductionmentioning

confidence: 99%

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A polynomial-time construction of a hitting set for read-once branching programs of width 3

Šíma,

Žák

2021

Preprint

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Recently, an interest in constructing pseudorandom or hitting set generators for restricted branching programs has increased, which is motivated by the fundamental issue of derandomizing space-bounded computations. Such constructions have been known only in the case of width 2 and in very restricted cases of bounded width. In this paper, we characterize the hitting sets for read-once branching programs of width 3 by a so-called richness condition. Namely, we show that such sets hit the class of read-once conjunctions of DNF and CNF (i.e. the weak richness). Moreover, we prove that any rich set extended with all strings within Hamming distance of 3 is a hitting set for read-once branching programs of width 3. Then, we show that any almost O(log n)-wise independent set satisfies the richness condition. By using such a set due to Alon et al. (1992) our result provides an explicit polynomial time construction of a hitting set for read-once branching programs of width 3 with acceptance probability ε > 5/6. We announced this result at conferences almost 10 years ago, including only proof sketches, which motivated a plenty of subsequent results on pseudorandom generators for restricted read-once branching programs. This paper contains our original detailed proof that has not been published yet.

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“…While the former class admits PRGs of poly-logarithmic seed length (see e.g. [ST19]), the most efficient PRG construction for the latter has seed length (1−o(1))•n [FSUV13]. Consequently, designing PRGs of seed length ≤ (1 − Ω(1)) • n can already be a challenge.…”

Section: Pseudorandom Generatorsmentioning

confidence: 99%

Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

Kabanets¹,

Koroth²,

Lu³

et al. 2020

Preprint

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The class FORMULA [s] • G consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class G. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n 1.99 ]• G, for classes G of functions with low communication complexity. Let R (k) (G) be the maximum k-party number-on-forehead randomized communication complexity of a function in G. Among other results, we show that:

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Cited by 18 publications

References 36 publications

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A polynomial-time construction of a hitting set for read-once branching programs of width 3

Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

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