All reports by Author Avishay Tal:

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TR18-112
| 5th June 2018
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Raghu Meka, Omer Reingold, Avishay Tal#### Pseudorandom Generators for Width-3 Branching Programs

Revisions: 1

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TR18-107
| 31st May 2018
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Ran Raz, Avishay Tal#### Oracle Separation of BQP and PH

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TR17-193
| 31st December 2017
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Oded Goldreich, Avishay Tal#### On Constant-Depth Canonical Boolean Circuits for Computing Multilinear Functions

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TR17-171
| 6th November 2017
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Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, Avishay Tal#### Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity

Revisions: 1

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TR17-148
| 6th October 2017
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Or Meir, Avishay Tal#### The Choice and Agreement Problems of a Random Function

Revisions: 3

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TR17-121
| 31st July 2017
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Sumegha Garg, Ran Raz, Avishay Tal#### Extractor-Based Time-Space Lower Bounds for Learning

Revisions: 1

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TR17-025
| 16th February 2017
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Pooya Hatami, Avishay Tal#### Pseudorandom Generators for Low-Sensitivity Functions

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TR16-181
| 15th November 2016
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Avishay Tal#### The Bipartite Formula Complexity of Inner-Product is Quadratic

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TR16-179
| 15th November 2016
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Avishay Tal#### Computing Requires Larger Formulas than Approximating

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TR16-113
| 22nd July 2016
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Gillat Kol, Ran Raz, Avishay Tal#### Time-Space Hardness of Learning Sparse Parities

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TR16-069
| 25th April 2016
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Parikshit Gopalan, Rocco Servedio, Avishay Tal, Avi Wigderson#### Degree and Sensitivity: tails of two distributions

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TR16-062
| 18th April 2016
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Avishay Tal#### On The Sensitivity Conjecture

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TR15-114
| 18th July 2015
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Avishay Tal#### #SAT Algorithms from Shrinkage

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TR15-079
| 7th May 2015
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Oded Goldreich, Avishay Tal#### Matrix Rigidity of Random Toeplitz Matrices

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TR14-174
| 14th December 2014
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Avishay Tal#### Tight bounds on The Fourier Spectrum of $AC^0$

Revisions: 2

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TR14-048
| 10th April 2014
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Avishay Tal#### Shrinkage of De Morgan Formulae from Quantum Query Complexity

Revisions: 1

Raghu Meka, Omer Reingold, Avishay Tal

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 >>>

Ran Raz, Avishay Tal

We present a distribution $D$ over inputs in $\{-1,1\}^{2N}$, such that:

(1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time $O(\log N)$, that distinguishes between $D$ and the uniform distribution with advantage $\Omega(1/\log N)$.

(2) No Boolean circuit of $\mathrm{quasipoly}(N)$ ...
more >>>

Oded Goldreich, Avishay Tal

We consider new complexity measures for the model of multilinear circuits with general multilinear gates introduced by Goldreich and Wigderson (ECCC, 2013).

These complexity measures are related to the size of canonical constant-depth Boolean circuits, which extend the definition of canonical depth-three Boolean circuits.

We obtain matching lower and upper ...
more >>>

Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, Avishay Tal

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 >>>

Or Meir, Avishay Tal

The direct-sum question is a classical question that asks whether

performing a task on $m$ independent inputs is $m$ times harder

than performing it on a single input. In order to study this question,

Beimel et. al (Computational Complexity 23(1), 2014) introduced the following related problems:

* The choice ... more >>>

Sumegha Garg, Ran Raz, Avishay Tal

A matrix $M: A \times X \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in X$ is chosen uniformly at random. A learner tries to learn $x$ from a stream of samples, $(a_1, b_1), (a_2, b_2) \ldots$, where for every $i$, $a_i \in A$ is chosen ... more >>>

Pooya Hatami, Avishay Tal

A Boolean function is said to have maximal sensitivity $s$ if $s$ is the largest number of Hamming neighbors of a point which differ from it in function value. We construct a pseudorandom generator with seed-length $2^{O(\sqrt{s})} \cdot \log(n)$ that fools Boolean functions on $n$ variables with maximal sensitivity at ... more >>>

Avishay Tal

A bipartite formula on binary variables $x_1, \ldots, x_n$ and $y_1, \ldots, y_n$ is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves may compute any function of either the $x$ or $y$ variables. We show that any bipartite formula for the Inner-Product ... more >>>

Avishay Tal

A de Morgan formula over Boolean variables $x_1, \ldots, x_n$ is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that ... more >>>

Gillat Kol, Ran Raz, Avishay Tal

We define a concept class ${\cal F}$ to be time-space hard (or memory-samples hard) if any learning algorithm for ${\cal F}$ requires either a memory of size super-linear in $n$ or a number of samples super-polynomial in $n$, where $n$ is the length of one sample.

A recent work shows ... more >>>

Parikshit Gopalan, Rocco Servedio, Avishay Tal, Avi Wigderson

The sensitivity of a Boolean function $f$ is the maximum, over all inputs $x$, of the number of sensitive coordinates of $x$ (namely the number of Hamming neighbors of $x$ with different $f$-value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-$s$ Boolean function ... more >>>

Avishay Tal

The sensitivity of a Boolean function $f:\{0,1\}^n \to \{0,1\}$ is the maximal number of neighbors a point in the Boolean hypercube has with different $f$-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the ... more >>>

Avishay Tal

We present a deterministic algorithm that counts the number of satisfying assignments for any de Morgan formula $F$ of size at most $n^{3-16\epsilon}$ in time $2^{n-n^{\epsilon}}\cdot \mathrm{poly}(n)$, for any small constant $\epsilon>0$. We do this by derandomizing the randomized algorithm mentioned by Komargodski et al. (FOCS, 2013) and Chen et ... more >>>

Oded Goldreich, Avishay Tal

We prove that random $n$-by-$n$ Toeplitz (alternatively Hankel) matrices over $GF(2)$ have rigidity $\Omega(\frac{n^3}{r^2\log n})$ for rank $r \ge \sqrt{n}$, with high probability. This improves, for $r = o(n/\log n \log\log n)$, over the $\Omega(\frac{n^2}{r} \cdot\log(\frac{n}{r}))$ bound that is known for many explicit matrices.

Our result implies that the explicit ... more >>>

Avishay Tal

We show that $AC^0$ circuits of depth $d$ and size $m$ have at most $2^{-\Omega(k/(\log m)^{d-1})}$ of their Fourier mass at level $k$ or above. Our proof builds on a previous result by H{\aa}stad (SICOMP, 2014) who proved this bound for the special case $k=n$. Our result is tight up ... more >>>

Avishay Tal

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 >>>