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Revision #1 to TR21-002 | 19th June 2021 13:06

#### Fooling Constant-Depth Threshold Circuits

Revision #1
Authors: Pooya Hatami, William Hoza, Avishay Tal, Roei Tell
Accepted on: 19th June 2021 13:07
Keywords:

Abstract:

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 $d\in\mathbb{N}$ and $n^{1+\delta}$ wires, where $\delta = 2^{-O(d)}$, using seed length $O(n^{1-\delta})$ and with error $2^{-n^{\delta}}$. This tightly matches the best known lower bounds for this circuit class. As a consequence of our result, all the known hardness for LTF circuits has now effectively been translated into pseudorandomness. This brings the extensive effort in the last decade to construct PRGs and deterministic circuit-analysis algorithms for this class to the point where any subsequent improvement would yield breakthrough lower bounds.

Our second contribution is a PRG for De Morgan formulas of size $s$ whose seed length is $s^{1/3+o(1)} \cdot \mathrm{polylog}(1/\epsilon)$ for error $\epsilon$. In particular, our PRG can fool formulas of sub-cubic size $s=n^{3-\Omega(1)}$ with an exponentially small error $\epsilon=\exp({-n^{\Omega(1)}})$. This significantly improves the inverse-polynomial error of the previous state-of-the-art for such formulas by Impagliazzo, Meka, and Zuckerman (FOCS 2012), and again tightly matches the best currently-known lower bounds for this class.

In both settings, a key ingredient in our constructions is a pseudorandom restriction procedure that has tiny failure probability, but simplifies the function to a non-natural "hybrid computational model" that combines several computational models. As part of our proofs we also construct "extremely low-error" PRGs for related circuit classes; for example, we construct a PRG for arbitrary functions of $s$ LTFs that can handle even the extreme setting of parameters $s=n/\mathrm{polylog}(n)$ and $\epsilon=2^{-n/\mathrm{polylog}(n)}$.

Changes to previous version:

Improved the PRG for TC0 to be strongly explicit, and deduced a lower bound for computing MCSP in TC0.

### Paper:

TR21-002 | 8th January 2021 22:04

#### Fooling Constant-Depth Threshold Circuits

TR21-002
Authors: Pooya Hatami, William Hoza, Avishay Tal, Roei Tell
Publication: 8th January 2021 23:01
Keywords:

Abstract:

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 $d\in\mathbb{N}$ and $n^{1+\delta}$ wires, where $\delta = 2^{-O(d)}$, using seed length $O(n^{1-\delta})$ and with error $2^{-n^{\delta}}$. This tightly matches the best known lower bounds for this circuit class. As a consequence of our result, all the known hardness for LTF circuits has now effectively been translated into pseudorandomness. This brings the extensive effort in the last decade to construct PRGs and deterministic circuit-analysis algorithms for this class to the point where any subsequent improvement would yield breakthrough lower bounds.

Our second contribution is a PRG for De Morgan formulas of size $s$ whose seed length is $s^{1/3+o(1)} \cdot \mathrm{polylog}(1/\epsilon)$ for error $\epsilon$. In particular, our PRG can fool formulas of sub-cubic size $s=n^{3-\Omega(1)}$ with an exponentially small error $\epsilon=\exp({-n^{\Omega(1)}})$. This significantly improves the inverse-polynomial error of the previous state-of-the-art for such formulas by Impagliazzo, Meka, and Zuckerman (FOCS 2012), and again tightly matches the best currently-known lower bounds for this class.

In both settings, a key ingredient in our constructions is a pseudorandom restriction procedure that has tiny failure probability, but simplifies the function to a non-natural "hybrid computational model" that combines several computational models. As part of our proofs we also construct "extremely low-error" PRGs for related circuit classes; for example, we construct a PRG for the class of functions computable by an arbitrary function of $s$ linear threshold functions that can handle even the extreme setting of parameters $s=n/\mathrm{polylog}(n)$ and $\epsilon=2^{-n/\mathrm{polylog}(n)}$.

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