Vikraman Arvind, K.V. Subrahmanyam, N. V. Vinodchandran

In this paper we study program checking (in the

sense of Blum and Kannan) using constant-depth circuits as

checkers. Our focus is on the number of queries made by the

checker to the program being checked and we term this as the

query complexity of the checker for the given ...
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Eric Allender, David Mix Barrington

The essential idea in the fast parallel computation of division and

related problems is that of Chinese remainder representation

(CRR) -- storing a number in the form of its residues modulo many

small primes. Integer division provides one of the few natural

examples of problems for which ...
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Eric Allender, David Mix Barrington, William Hesse

Integer division has been known to lie in P-uniform TC^0 since

the mid-1980's, and recently this was improved to DLOG-uniform

TC^0. At the time that the results in this paper were proved and

submitted for conference presentation, it was unknown whether division

lay in DLOGTIME-uniform TC^0 (also known as ...
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Alexander Healy

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in AC^0[mod 2]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC^1. For example, we obtain the following results:

... more >>>Dan Gutfreund, Alexander Healy, Tali Kaufman, Guy Rothblum

Program checking, program self-correcting and program self-testing

were pioneered by [Blum and Kannan] and [Blum, Luby and Rubinfeld] in

the mid eighties as a new way to gain confidence in software, by

considering program correctness on an input by input basis rather than

full program verification. Work in ...
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Paul Beame, Trinh Huynh

We prove an n^{Omega(1)}/2^{O(k)} lower bound on the randomized k-party communication complexity of read-once depth 4 AC^0 functions in the number-on-forehead (NOF) model for O(log n) players. These are the first non-trivial lower bounds for general NOF multiparty communication complexity for any AC^0 function for omega(log log n) players. For ... more >>>

Vikraman Arvind, Yadu Vasudev

Given two $n$-variable boolean functions $f$ and $g$, we study the problem of computing an $\varepsilon$-approximate isomorphism between them. I.e.\ a permutation $\pi$ of the $n$ variables such that $f(x_1,x_2,\ldots,x_n)$ and $g(x_{\pi(1)},x_{\pi(2)},\ldots,x_{\pi(n)})$ differ on at most an $\varepsilon$ fraction of all boolean inputs $\{0,1\}^n$. We give a randomized $2^{O(\sqrt{n}\log(n)^{O(1)})}$ algorithm ... more >>>

Matthew Anderson, Dieter van Melkebeek, Nicole Schweikardt, Luc Segoufin

We study the locality of an extension of first-order logic that captures graph queries computable in AC$^0$, i.e., by families of polynomial-size constant-depth circuits. The extension considers first-order formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the ... more >>>

Chris Beck, Russell Impagliazzo, Shachar Lovett

There has been considerable interest lately in the complexity of distributions. Recently, Lovett and Viola (CCC 2011) showed that the statistical distance between a uniform distribution over a good code, and any distribution which can be efficiently sampled by a small bounded-depth AC0 circuit, is inverse-polynomially close to one. That ... more >>>

Swastik Kopparty, Srikanth Srinivasan

In this paper, we introduce and develop the method of certifying polynomials for proving $\mathrm{AC}^0[\oplus]$ circuit lower bounds.

We use this method to show that Approximate Majority cannot be computed by $\mathrm{AC}^0[\oplus]$ circuits of size $n^{1+o(1)}$. This implies a separation between the power of $\mathrm{AC}^0[\oplus]$ circuits of near-linear size and ... more >>>

Arkadev Chattopadhyay, Rahul Santhanam

We formulate a new connection between instance compressibility \cite{Harnik-Naor10}), where the compressor uses circuits from a class $\C$, and correlation with

circuits in $\C$. We use this connection to prove the first lower bounds

on general probabilistic multi-round instance compression. We show that there

is no

probabilistic multi-round ...
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Yuan Li, Alexander Razborov, Benjamin Rossman

Let $P$ be a fixed graph (hereafter called a ``pattern''), and let

$Subgraph(P)$ denote the problem of deciding whether a given graph $G$

contains a subgraph isomorphic to $P$. We are interested in

$AC^0$-complexity of this problem, determined by the smallest possible exponent

$C(P)$ for which $Subgraph(P)$ possesses bounded-depth circuits ...
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Benny Applebaum

Constant parallel-time cryptography allows to perform complex cryptographic tasks at an ultimate level of parallelism, namely, by local functions that each of their output bits depend on a constant number of input bits. A natural way to obtain local cryptographic constructions is to use \emph{random local functions} in which each ... more >>>

Benny Applebaum, Shachar Lovett

Suppose that you have $n$ truly random bits $x=(x_1,\ldots,x_n)$ and you wish to use them to generate $m\gg n$ pseudorandom bits $y=(y_1,\ldots, y_m)$ using a local mapping, i.e., each $y_i$ should depend on at most $d=O(1)$ bits of $x$. In the polynomial regime of $m=n^s$, $s>1$, the only known solution, ... more >>>

Roei Tell

Goldreich and Wigderson (STOC 2014) initiated a study of quantified derandomization, which is a relaxed derandomization problem: For a circuit class $\mathcal{C}$ and a parameter $B=B(n)$, the problem is to decide whether a circuit $C\in\mathcal{C}$ rejects all of its inputs, or accepts all but $B(n)$ of its inputs.

In ... more >>>

Valentine Kabanets, Zhenjian Lu

A polynomial threshold function (PTF) is defined as the sign of a polynomial $p\colon\bool^n\to\mathbb{R}$. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.

Satisfiability (#SAT). We give the first zero-error randomized algorithm ... more >>>

Alexander A. Sherstov, Pei Wu

The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum ... more >>>

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function $f$ can be computed by a Boolean circuit of size at most $\theta$, for a given parameter $\theta$. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a ... more >>>

Alexander A. Sherstov

The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that approximates $f$ pointwise: $|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we construct CNF and DNF formulas of polynomial size with approximate degree $\Omega(n^{1-\delta}),$ essentially matching the trivial upper bound of $n.$ This ... more >>>

Bruno Pasqualotto Cavalar, Igor Carboni Oliveira

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:

- For every $k \geq 1$, there is a monotone function in ${\rm AC^0}$ (constant-depth poly-size circuits) that requires monotone circuits of depth $\Omega(\log^k n)$. This significantly extends a classical result of Okol'nishnikova (1982) and Ajtai ... more >>>