Towards the Pseudorandomness of Expander Random Walks for Read-Once ACC0 circuits

TL;DR

This paper investigates the pseudorandomness of expander graph-based random walks against complex asymmetric circuits like $ ext{ACC}^0$, showing they can fool certain two-layered $ ext{MOD}[k]$ circuits but not some $ ext{TC}^0$ circuits.

Contribution

It advances understanding of which complex circuits can be fooled by expander random walks, especially for asymmetric circuit classes like $ ext{ACC}^0$ and $ ext{MOD}[k]$.

Findings

Expander random walks fool certain two-layered $ ext{MOD}[k]$ circuits with error $O( extlambda)$.

Highly asymmetric circuits with complex Fourier characters can be fooled by expander walks.

Some $ ext{TC}^0$ circuits are not fooled by expander random walks, setting limits on their pseudorandomness.

Abstract

Expander graphs are among the most useful combinatorial objects in theoretical computer science. A line of work studies random walks on expander graphs for their pseudorandomness against various classes of test functions, including symmetric functions, read-only branching programs, permutation branching programs, and $\mathrm{AC}^0$ circuits. The promising results of pseudorandomness of expander random walks against $\mathrm{AC}^0$ circuits indicate a robustness of expander random walks beyond symmetric functions, motivating the question of whether expander random walks can fool more robust \emph{asymmetric} complexity classes, such as $\mathrm{ACC}^0$. In this work, we make progress towards this question by considering certain two-layered circuit compositions of $\mathrm{MOD}[k]$ gates, where we show that these family of circuits are fooled by expander random walks with total variation distance error $O(\lambda)$, where $\lambda$ is the second largest eigenvalue of the underlying expander graph. For $k\geq 3$, these circuits can be highly asymmetric with complicated Fourier characters. In this context, our work takes a step in the direction of fooling more complex asymmetric circuits. Separately, drawing from the learning-theory literature, we construct an explicit threshold circuit in the circuit family $\mathrm{TC}^0$, and show that it is \emph{not} fooled by expander random walk, providing an upper bound on the set of functions fooled by expander random walks.