A Fixed-Depth Size-Hierarchy Theorem for $\mathrm{AC}^0[\oplus]$ via the Coin Problem

Limaye, Nutan; Sreenivasaiah, Karteek; Srinivasan, Srikanth; Tripathi, Utkarsh; Venkitesh, S.

doi:10.1137/19m1276467

Cited by 3 publications

(6 citation statements)

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“…[3,46,47,39,12,15]) and for various reasons such as understanding the power of randomness in bounded-depth circuits, the limitations of blackbox hardness ampli cation, and devising pseudorandom generators for bounded-width branching programs. More recently, Limaye et al [31] proved optimal lower bounds on the size of AC 0 [⊕] 3 circuits solving the 𝛿-coin problem with constant error, strengthening an earlier lower bound of Shaltiel and Viola [39]. This led to the rst class of explicit functions for which we have tight (up to polynomial factors) AC 0 [⊕] lower bounds.…”

Section: Applicationsmentioning

confidence: 56%

“…In earlier work [31], we showed that this was tight for constant 𝜀. That is, we showed that any polynomial 𝑃 that solves the 𝛿-coin problem with error at most 1/10 (say) must have degree Ω(1/𝛿).…”

Section: E F I N I T I O Nmentioning

confidence: 85%

“…A key result in the lower bound of Limaye et al [31] was a tight lower bound on the degree of any polynomial 𝑃 ∈ F[𝑥 1 , . .…”

Section: Applicationsmentioning

confidence: 99%

“…Using the robust Hegedűs lemma, we are able to strengthen the degree lower bound of [31] to a tight degree lower bound for all errors. Speci cally, we show that over any eld F of xed positive characteristic 𝑝, any polynomial 𝑃 that solves the 𝛿-coin problem with error 𝜀 must have degree Ω( 1𝛿 log(1/𝜀)), which is tight for all 𝛿 and 𝜀. where we think of polynomials as algorithms and degree as a measure of e ciency.…”

Section: Applicationsmentioning

confidence: 99%

“…(This de nition is sometimes [31] stated in terms of the distributions 𝜇 (1/2)−𝛿 and 𝜇 (1/2)+𝛿 . This is essentially equivalent to the de nition above.…”

Section: E F I N I T I O Nmentioning

confidence: 99%

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A Robust Version of Heged\H{u}s's Lemma, with Applications

Srinivasan¹

2023

TheoretiCS

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Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.

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Section: Applicationsmentioning

confidence: 56%

Section: E F I N I T I O Nmentioning

confidence: 85%

“…A key result in the lower bound of Limaye et al [31] was a tight lower bound on the degree of any polynomial 𝑃 ∈ F[𝑥 1 , . .…”

Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

“…(This de nition is sometimes [31] stated in terms of the distributions 𝜇 (1/2)−𝛿 and 𝜇 (1/2)+𝛿 . This is essentially equivalent to the de nition above.…”

Section: E F I N I T I O Nmentioning

confidence: 99%

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