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Steinke, Thomas; Vadhan, Salil; Wan, Andrew

doi:10.4086/toc.2017.v013a012

Cited by 12 publications

(4 citation statements)

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“…A preliminary version of this article appeared as extended abstracts [17,18] including only proof sketches, where our result was formulated for acceptance probability ε > 11/12. Since then a plenty of results on pseudorandom generators for restricted 1-branching programs [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] have been achieved which were motivated and/or follows our study referring to our result; see e.g. the paper [36] for a current survey of the newest achievements along this direction.…”

Section: Introductionmentioning

confidence: 77%

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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Section: Introductionmentioning

confidence: 77%

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

Šíma,

Žák

2021

Preprint

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“…One thing to note about the above proof is that the coin theorem we used of [22] was suboptimal in the range ε = n −ω (1) , but still implied the optimal level 1 bound of [23]. Using Proposition 2.1, we can also improve the [22] coin theorem for small ε by setting ε 0 = 1/n in the following corollary:…”

Section: Definition 13 ([20]mentioning

confidence: 92%

“…in 2014 to a corresponding level 1 Fourier bound for (non-necessarily monotone) constant-width read-once branching programs. Applying Corollary 2.3, we see that these latter results can in fact be derived (up to constant factors) using Steinberger's coin theorem as a black box.Corollary 2.4 ([22,23]). Let f : {−1, 1} n → {−1, 1} be computable by a width-w read-once branching program.…”

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confidence: 99%

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Agrawal¹

2020

Chicago J. of Theoretical Comp. Sci.

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In this note we compare two measures of the complexity of a class F of Boolean functions studied in (unconditional) pseudorandomness: F's ability to distinguish between biased and uniform coins (the coin problem), and the norms of the different levels of the Fourier expansion of functions in F (the Fourier growth). We show that for coins with low bias ε = o(1/n), a function's distinguishing advantage in the coin problem is essentially equivalent to ε times the sum of its level 1 Fourier coefficients, which in particular shows that known level 1 and total influence bounds for some classes of interest (such as constant-width read-once branching programs) in fact follow as a black-box from the corresponding coin theorems, thereby simplifying the proofs of some known results in the literature. For higher levels, it is well-known that Fourier growth bounds on all levels of the Fourier spectrum imply coin theorems, even for large ε, and we discuss here the possibility of a converse.

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“…Even prior to Tal's work, the ℓ-Fourier weight of decision trees was studied for ℓ = 1 by O'Donnell and Servedio [17], who proved the tight O( √ d) bound and used it to give a polynomial-time learning algorithm for monotone decision trees. Fourier weight has been studied for various other classes of Boolean functions, including boundeddepth circuits, branching programs, low-degree polynomials over finite fields, and functions with bounded sensitivity; see the recent papers [12,20,21,10,9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

An Optimal Separation of Randomized and Quantum Query Complexity

Sherstov

Storozhenko

2020

Preprint

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We prove that for every decision tree, the absolute values of the Fourier coefficients of given order ℓ 1 sum to at most cwhere n is the number of variables, d is the tree depth, and c > 0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with ℓ, becoming trivial already at ℓ = √ d. As an application, we obtain, for any positive integer k, a partial Boolean function on n bits that has bounded-error quantum query complexity at most ⌈k/2⌉ and randomized query complexity Ω(n 1−1/k ). This separation of boundederror quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015). Prior to our work, the best known separation was polynomially weaker: O(1) versus n 2/3−ε for any ε > 0 (Tal, FOCS 2020).

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Cited by 12 publications

References 50 publications

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

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

Untitled

An Optimal Separation of Randomized and Quantum Query Complexity

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