Towards the Pseudorandomness of Expander Random Walks for Read-Once ACC0 circuits
Emile Anand
TL;DR
The paper advances the understanding of expander random walks as pseudorandom generators against asymmetric circuits by proving that two-layer compositions of MOD[$k$] gates (with $k\ge 3$) and their variants are fooled with total-variation error $O(\lambda)$, where $\lambda$ is the expander's second eigenvalue. It further generalizes to depth-2 compositions of varying symmetric functions and an AND gate, establishing a broader pseudorandomness window within ACC$^0$. In contrast, the authors construct an explicit TC$^0$ threshold circuit that is not fooled by expander walks, demonstrating limitations and outlining a boundary for the approach. Collectively, the results push toward understanding which constant-depth, read-many/asymmetric circuits can be derandomized by expander-based PRGs, while clarifying that some natural higher-order thresholds remain distinguishable from uniform walks. The work leverages tools such as alternate conditioning and the notion of Maximum Pseudorandom Variation to handle dependencies in expander-walk inputs and employs spectral properties to quantify the fooling error.
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 $O(λ)$, where $λ$ 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.