All reports by Author William Hoza:

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TR23-019
| 2nd March 2023
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Pooya Hatami, William Hoza#### Theory of Unconditional Pseudorandom Generators

Revisions: 2

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TR22-121
| 27th August 2022
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William Hoza#### Recent Progress on Derandomizing Space-Bounded Computation

Revisions: 1

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TR22-087
| 8th June 2022
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Pooya Hatami, William Hoza, Avishay Tal, Roei Tell#### Depth-$d$ Threshold Circuits vs. Depth-$(d + 1)$ AND-OR Trees

Revisions: 1

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TR21-048
| 27th March 2021
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William Hoza#### Better Pseudodistributions and Derandomization for Space-Bounded Computation

Revisions: 1

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TR21-002
| 8th January 2021
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Pooya Hatami, William Hoza, Avishay Tal, Roei Tell#### Fooling Constant-Depth Threshold Circuits

Revisions: 1

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TR20-138
| 9th September 2020
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William Hoza, Edward Pyne, Salil Vadhan#### Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

Revisions: 1

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TR20-016
| 17th February 2020
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Kuan Cheng, William Hoza#### Hitting Sets Give Two-Sided Derandomization of Small Space

Revisions: 1

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TR19-149
| 4th November 2019
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Dean Doron, Pooya Hatami, William Hoza#### Log-Seed Pseudorandom Generators via Iterated Restrictions

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TR18-183
| 5th November 2018
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Dean Doron, Pooya Hatami, William Hoza#### Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

Revisions: 2

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TR18-063
| 5th April 2018
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William Hoza, David Zuckerman#### Simple Optimal Hitting Sets for Small-Success $\mathbf{RL}$

Revisions: 1

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TR16-172
| 3rd November 2016
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William Hoza, Adam Klivans#### Preserving Randomness for Adaptive Algorithms

Revisions: 4

Pooya Hatami, William Hoza

This is a survey of unconditional *pseudorandom generators* (PRGs). A PRG uses a short, truly random seed to generate a long, "pseudorandom" sequence of bits. To be more specific, for each restricted model of computation (e.g., bounded-depth circuits or read-once branching programs), we would like to design a PRG that ... more >>>

William Hoza

Is randomness ever necessary for space-efficient computation? It is commonly conjectured that L = BPL, meaning that halting decision algorithms can always be derandomized without increasing their space complexity by more than a constant factor. In the past few years (say, from 2017 to 2022), there has been some exciting ... more >>>

Pooya Hatami, William Hoza, Avishay Tal, Roei Tell

For $n \in \mathbb{N}$ and $d = o(\log \log n)$, we prove that there is a Boolean function $F$ on $n$ bits and a value $\gamma = 2^{-\Theta(d)}$ such that $F$ can be computed by a uniform depth-$(d + 1)$ $\text{AC}^0$ circuit with $O(n)$ wires, but $F$ cannot be computed ... more >>>

William Hoza

Three decades ago, Nisan constructed an explicit pseudorandom generator (PRG) that fools width-$n$ length-$n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2 n + \log n \cdot \log(1/\varepsilon))$ (Combinatorica 1992). Nisan's generator remains the best explicit PRG known for this important model of computation. However, a recent ... more >>>

Pooya Hatami, William Hoza, Avishay Tal, Roei Tell

We present new constructions of pseudorandom generators (PRGs) for two of the most widely-studied non-uniform circuit classes in complexity theory. Our main result is a construction of the first non-trivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth $d\in\mathbb{N}$ ... more >>>

William Hoza, Edward Pyne, Salil Vadhan

We prove that the Impagliazzo-Nisan-Wigderson (STOC 1994) pseudorandom generator (PRG) fools ordered (read-once) permutation branching programs of unbounded width with a seed length of $\widetilde{O}(\log d + \log n \cdot \log(1/\varepsilon))$, assuming the program has only one accepting vertex in the final layer. Here, $n$ is the length of the ... more >>>

Kuan Cheng, William Hoza

A hitting set is a "one-sided" variant of a pseudorandom generator (PRG), naturally suited to derandomizing algorithms that have one-sided error. We study the problem of using a given hitting set to derandomize algorithms that have two-sided error, focusing on space-bounded algorithms. For our first result, we show that if ... more >>>

Dean Doron, Pooya Hatami, William Hoza

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The ``iterated restrictions'' approach, pioneered by Ajtai and Wigderson [AW89], has provided PRGs with seed length $\mathrm{polylog} n$ or even $\tilde{O}(\log n)$ for several restricted models of computation. Can this approach ever achieve the optimal seed ... more >>>

Dean Doron, Pooya Hatami, William Hoza

We give an explicit pseudorandom generator (PRG) for constant-depth read-once formulas over the basis $\{\wedge, \vee, \neg\}$ with unbounded fan-in. The seed length of our PRG is $\widetilde{O}(\log(n/\varepsilon))$. Previously, PRGs with near-optimal seed length were known only for the depth-2 case (Gopalan et al. FOCS '12). For a constant depth ... more >>>

William Hoza, David Zuckerman

We give a simple explicit hitting set generator for read-once branching programs of width $w$ and length $r$ with known variable order. Our generator has seed length $O\left(\frac{\log(wr) \log r}{\max\{1, \log \log w - \log \log r\}} + \log(1/\varepsilon)\right)$. This seed length improves on recent work by Braverman, Cohen, and ... more >>>

William Hoza, Adam Klivans

We introduce the concept of a randomness steward, a tool for saving random bits when executing a randomized estimation algorithm $\mathrm{Est}$ on many adaptively chosen inputs. For each execution, the chosen input to $\mathrm{Est}$ remains hidden from the steward, but the steward chooses the randomness of $\mathrm{Est}$ and, crucially, is ... more >>>