Volume 18 (2022) Article 21 pp. 1-32
CCC 2020 Special Issue
Hitting Sets Give Two-Sided Derandomization of Small Space
by
Revised: June 21, 2022
Published: September 20, 2022
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Keywords: derandomization, pseudorandomness, hitting set, branching programs, space complexity
ACM Classification: F.1.3, F.1.2
AMS Classification: 68Q17, 68Q25

Abstract: [Plain Text Version]

$\newcommand{\EE}{\mathbb{E}}$

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 there is a log-space hitting set for polynomial-width read-once branching programs (ROBPs), then not only does $\mathbf{L} = \mathbf{RL}$ hold, but $\mathbf{L} = \mathbf{BPL}$ as well. This answers a question raised by Hoza and Zuckerman (SICOMP 2020).

Next, we consider constant-width ROBPs. We show that if there are log-space hitting sets for constant-width ROBPs, then given black-box access to a constant-width ROBP $f$, it is possible to deterministically estimate $\EE[f]$ to within $\pm \epsilon$ in space $O(\log(n/\epsilon))$. Unconditionally, we give a deterministic algorithm for this problem with space complexity $O(\log^2 n + \log(1/\epsilon))$, slightly improving over previous work.

Finally, we investigate the limits of this line of work. Perhaps the strongest reduction along these lines one could hope for would say that for every explicit hitting set generator, there is an explicit PRG with similar parameters. In the setting of constant-width ROBPs over a large alphabet, we prove that establishing such a strong reduction is at least as difficult as constructing a good PRG outright. Quantitatively, we prove that if the strong reduction holds, then for every constant $\alpha > 0$, there is an explicit PRG for constant-width ROBPs with seed length $O(\log^{1 + \alpha} n)$. Along the way, unconditionally, we construct an improved hitting set for ROBPs over a large alphabet.

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A conference version of this paper appeared in the Proceedings of the 35th Computational Complexity Conference, 2020.