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REPORTS > KEYWORD > RANDOM RESTRICTIONS:
Reports tagged with random restrictions:
TR98-036 | 11th June 1998
Vince Grolmusz, Gábor Tardos

#### Lower Bounds for (MOD p -- MOD m) Circuits

Modular gates are known to be immune for the random
restriction techniques of Ajtai; Furst, Saxe, Sipser; and Yao and
Hastad. We demonstrate here a random clustering technique which
overcomes this difficulty and is capable to prove generalizations of
several known modular circuit lower bounds of Barrington, Straubing,
Therien; Krause ... more >>>

TR12-057 | 7th May 2012
Russell Impagliazzo, Raghu Meka, David Zuckerman

#### Pseudorandomness from Shrinkage

Revisions: 2

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models ... more >>>

TR12-123 | 28th September 2012
Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, Salil Vadhan

#### Better pseudorandom generators from milder pseudorandom restrictions

We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs, all of which achieve near optimal seed-length even in ... more >>>

TR13-024 | 7th February 2013
Valentine Kabanets, Antonina Kolokolova

#### Compression of Boolean Functions

We consider the problem of compression for easy'' Boolean functions: given the truth table of an $n$-variate Boolean function $f$ computable by some \emph{unknown small circuit} from a \emph{known class} of circuits, find in deterministic time $\poly(2^n)$ a circuit $C$ (no restriction on the type of $C$) computing $f$ so ... more >>>

TR13-057 | 5th April 2013
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, David Zuckerman

#### Mining Circuit Lower Bound Proofs for Meta-Algorithms

We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for easy'' Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an $n$-variate Boolean function $f$ computable by some unknown small circuit ... more >>>

TR13-058 | 5th April 2013
Ilan Komargodski, Ran Raz, Avishay Tal

#### Improved Average-Case Lower Bounds for DeMorgan Formula Size

Revisions: 2

We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every deMorgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013).

Our technical contributions include a theorem that shows that the expected ... more >>>

TR13-150 | 4th November 2013
Ruiwen Chen, Valentine Kabanets, Nitin Saurabh

#### An Improved Deterministic #SAT Algorithm for Small De Morgan Formulas

We give a deterministic #SAT algorithm for de Morgan formulas of size up to $n^{2.63}$, which runs in time $2^{n-n^{\Omega(1)}}$. This improves upon the deterministic #SAT algorithm of \cite{CKKSZ13}, which has similar running time but works only for formulas of size less than $n^{2.5}$.

Our new algorithm is based on ... more >>>

TR13-163 | 28th November 2013
Russell Impagliazzo, Valentine Kabanets

#### Fourier Concentration from Shrinkage

Revisions: 2

For Boolean functions computed by de Morgan formulas of subquadratic size or read-once de Morgan formulas, we prove a sharp concentration of the Fourier mass on small-degree'' coefficients. For a Boolean function $f:\{0,1\}^n\to\{1,-1\}$ computable by a de Morgan formula of size $s$, we show that
\[
\sum_{A\subseteq [n]\; :\; |A|>s^{1/\Gamma ... more >>>

TR14-048 | 10th April 2014
Avishay Tal

#### Shrinkage of De Morgan Formulae from Quantum Query Complexity

Revisions: 1

We give a new and improved proof that the shrinkage exponent of De Morgan formulae is $2$. Namely, we show that for any Boolean function $f: \{-1,1\}^n \to \{-1,1\}$, setting each variable out of $x_1, \ldots, x_n$ with probability $1-p$ to a randomly chosen constant, reduces the expected formula size ... more >>>

TR15-188 | 24th November 2015
Daniel Kane, Ryan Williams

#### Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for ... more >>>

TR17-026 | 17th February 2017
Valentine Kabanets, Daniel Kane, Zhenjian Lu

#### A Polynomial Restriction Lemma with Applications

A polynomial threshold function (PTF) of degree $d$ is a boolean function of the form $f=\mathrm{sgn}(p)$, where $p$ is a degree-$d$ polynomial, and $\mathrm{sgn}$ is the sign function. The main result of the paper is an almost optimal bound on the probability that a random restriction of a PTF is ... more >>>

TR17-171 | 6th November 2017
Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, Avishay Tal

#### Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity

Revisions: 1

We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS'12) where they required seed length $n^{1/2+o(1)}$.

A central ingredient in our work ... more >>>

TR18-112 | 5th June 2018
Raghu Meka, Omer Reingold, Avishay Tal

#### Pseudorandom Generators for Width-3 Branching Programs

Revisions: 1

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work ... more >>>

TR18-160 | 12th September 2018
Anna Gal, Avishay Tal, Adrian Trejo Nuñez

#### Cubic Formula Size Lower Bounds Based on Compositions with Majority

We define new functions based on the Andreev function and prove that they require $n^{3}/polylog(n)$ formula size to compute. The functions we consider are generalizations of the Andreev function using compositions with the majority function. Our arguments apply to composing a hard function with any function that agrees with the ... more >>>

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