Electronic Colloquium on Computational Complexity
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Reports tagged with constant-depth circuits:
TR98-078 | 1st December 1998
Vikraman Arvind, K.V. Subrahmanyam, Vinodchandran Variyam

The Query Complexity of Program Checking by Constant-Depth Circuits

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 ... more >>>

TR00-065 | 7th September 2000
Eric Allender, David Mix Barrington

Uniform Circuits for Division: Consequences and Problems

Comments: 2

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 ... more >>>

TR01-033 | 27th April 2001
Eric Allender, David Mix Barrington, William Hesse

Uniform Circuits for Division: Consequences and Problems

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 ... more >>>

TR06-058 | 25th April 2006
Alexander Healy

Randomness-Efficient Sampling within NC^1

Revisions: 1

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 >>>

TR07-047 | 15th May 2007
Dan Gutfreund, Alexander Healy, Tali Kaufman, Guy Rothblum

A (De)constructive Approach to Program Checking

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 ... more >>>

TR08-082 | 11th September 2008
Paul Beame, Trinh Huynh

Multiparty Communication Complexity and Threshold Circuit Size of AC^0

Revisions: 2

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 >>>

TR11-137 | 14th October 2011
Vikraman Arvind, Yadu Vasudev

Isomorphism Testing of Boolean Functions Computable by Constant Depth Circuits

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 >>>

TR11-158 | 25th November 2011
Matthew Anderson, Dieter van Melkebeek, Nicole Schweikardt, Luc Segoufin

Locality from Circuit Lower Bounds

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 >>>

TR12-042 | 17th April 2012
Chris Beck, Russell Impagliazzo, Shachar Lovett

Large Deviation Bounds for Decision Trees and Sampling Lower Bounds for AC0-circuits

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 >>>

TR12-102 | 16th August 2012
Swastik Kopparty, Srikanth Srinivasan

Certifying Polynomials for $\mathrm{AC}^0[\oplus]$ circuits, with applications

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 >>>

TR12-108 | 4th September 2012
Arkadev Chattopadhyay, Rahul Santhanam

Lower Bounds on Interactive Compressibility by Constant-Depth Circuits

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 ... more >>>

TR14-145 | 4th November 2014
Yuan Li, Alexander Razborov, Benjamin Rossman

On the $AC^0$ Complexity of Subgraph Isomorphism

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 ... more >>>

TR15-027 | 25th February 2015
Benny Applebaum

Cryptographic Hardness of Random Local Functions -- Survey

Revisions: 1

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 >>>

TR15-172 | 3rd November 2015
Benny Applebaum, Shachar Lovett

Algebraic Attacks against Random Local Functions and Their Countermeasures

Revisions: 1

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 >>>

TR16-191 | 24th November 2016
Roei Tell

Improved Bounds for Quantified Derandomization of Constant-Depth Circuits and Polynomials

Revisions: 1

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 >>>

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