Improved pseudorandomness for unordered branching programs through local monotonicity

Chattopadhyay, Eshan; Hatami, Pooya; Reingold, Omer; Tal, Avishay

doi:10.1145/3188745.3188800

Cited by 17 publications

(24 citation statements)

References 27 publications

(31 reference statements)

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“…The following is a restatement of a result from [CHRT17]. We give its proof for completeness in Appendix A.1.…”

Section: Branching Programsmentioning

confidence: 88%

“…, f m are computed by constant-width ROBPs. We rely on the previous work of Chattopadhyay, Hatami, Reingold, Tal [CHRT17]. They constructed PRGs for constant-width length-n ROBPs with seed-length poly log(n).…”

Section: Pseudorandom Restrictions For the Xor Of Short Width-3 Robpsmentioning

confidence: 99%

“…In the following, we say that a read-once branching program B is δ-reachable if for all reachable vertices v in B we have Next, we reduce the task of fooling ROBPs with at most ℓ-colliding layers to the task of fooling δ-reachable ROBPs. The reduction is similar to that in [CHRT17]. The main difference is that we simulate a ROBP with width w by a δ-reachable ROBP of width w + 1 by adding a new sink state that should be thought of as "immediate stop".…”

Section: Prgs For Robps With Few Colliding Layersmentioning

confidence: 99%

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Pseudorandom generators for width-3 branching programs

Meka

Reingold

Tal

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We construct pseudorandom generators of seed length O(log(n)·log(1/ε)) that ε-fool ordered read-once branching programs (ROBPs) of width 3 and length n. For unordered ROBPs, we construct pseudorandom generators with seed length O(log(n) · poly(1/ε)). This is the first improvement for pseudorandom generators fooling width 3 ROBPs since the work of Nisan [Nis92].Our constructions are based on the "iterated milder restrictions" approach of [GMR + 12] (which further extends the Ajtai-Wigderson framework [AW85]), combined with the INW-generator [INW94] at the last step (as analyzed by [BRRY14]). For the unordered case we combine iterated milder restrictions with the generator of [CHHL18].Two conceptual ideas that play an important role in our analysis are: *

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“…The following is a restatement of a result from [CHRT17]. We give its proof for completeness in Appendix A.1.…”

Section: Branching Programsmentioning

confidence: 88%

Section: Pseudorandom Restrictions For the Xor Of Short Width-3 Robpsmentioning

confidence: 99%

Section: Prgs For Robps With Few Colliding Layersmentioning

confidence: 99%

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Pseudorandom generators for width-3 branching programs

Meka

Reingold

Tal

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…Ajtai and Wigderson [1] proposed such a PRG for low-depth circuits. Much follow-up work has been based on this framework to build PRGs for various classes of functions including low-depth circuits, branching programs, low-sensitivity functions [23,8,20,6,10], and a major component of the analysis is proving that the derandomized random restrictions work.…”

Section: Prg For Functions Which Simplify Under Random Restrictionmentioning

confidence: 99%

Untitled

Chattopadhyay¹,

Hatami²,

Hosseini³

et al. 2019

Theory of Comput.

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We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [−1, 1] n. Next, we use a fractional pseudorandom generator as steps of a random walk in [−1, 1] n that converges to {−1, 1} n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded-depth circuits.

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“…However, we can still make use of one to replace otherwise naive bounds on Fourier mass in our argument. By incorporating the level-k Fourier mass bound for constant-width branching programs derived in [CHRT17] into our approach, we get the following improvement on Theorem 2.1 in the constant-width case.…”

Section: The Constant-width Casementioning

confidence: 99%

Pseudorandom Generators for Read-Once Branching Programs, in Any Order

Forbes

Kelley

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL = L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan [Nis92], pseudorandom generators with seed-length O(log 2 n) were constructed (see also [INW94,GR14]). Unfortunately, improving on this seed-length in general has proven challenging and seems to require new ideas.A recent line of inquiry (e.g., [BV10, GMR + 12, IMZ12, RSV13, SVW14, HLV17, LV17, CHRT17]) has suggested focusing on a particular limitation of the existing PRGs ([Nis92, INW94, GR14]), which is that they only fool roBPs when the variables are read in a particular known order, such as x 1 < · · · < x n . In comparison, existentially one can obtain logarithmic seed-length for fooling the set of polynomial-size roBPs that read the variables under any fixed unknown permutation x π(1) < · · · < x π(n) . While recent works have established novel PRGs in this setting for subclasses of roBPs, there were no known n o(1) seed-length explicit PRGs for general polynomial-size roBPs in this setting.In this work, we follow the "bounded independence plus noise" paradigm of Haramaty, Lee and Viola [HLV17, LV17], and give an improved analysis in the general roBP unknown-order setting. With this analysis we obtain an explicit PRG with seed-length O(log 3 n) for polynomialsize roBPs reading their bits in an unknown order. Plugging in a recent Fourier tail bound of Chattopadhyay, Hatami, Reingold, and Tal [CHRT17], we can obtain a O(log 2 n) seed-length when the roBP is of constant width. *

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Improved pseudorandomness for unordered branching programs through local monotonicity

Cited by 17 publications

References 27 publications

Pseudorandom generators for width-3 branching programs

Pseudorandom generators for width-3 branching programs

Untitled

Pseudorandom Generators for Read-Once Branching Programs, in Any Order

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