Near-optimal pseudorandom generators for constant-depth read-once formulas

We give an explicit pseudorandom generator (PRG) for read-once AC⁰, i.e., constant-depth read-once formulas over the basis {∧, ∨, ¬} with unbounded fan-in. The seed length of our PRG is O^e(log(n/ε)). Previously, PRGs with near-optimal seed length were known only for the depth-2 case [22]. For a constant depth d > 2, the best prior PRG is a recent construction by Forbes and Kelley with seed length O^e(log² n + log n log(1/ε)) for the more general model of constant-width read-once branching programs with arbitrary variable order [17]. Looking beyond read-once AC⁰, we also show that our PRG fools read-once AC⁰[⊕] with seed length O^e(t + log(n/ε)), where t is the number of parity gates in the formula. Our construction follows Ajtai and Wigderson’s approach of iterated pseudorandom restrictions [1]. We assume by recursion that we already have a PRG for depth-d AC⁰ formulas. To fool depth-(d+ 1) AC⁰ formulas, we use the given PRG, combined with a small-bias distribution and almost k-wise independence, to sample a pseudorandom restriction. The analysis of Forbes and Kelley [17] shows that our restriction approximately preserves the expectation of the formula. The crux of our work is showing that after poly(log log n) independent applications of our pseudorandom restriction, the formula simplifies in the sense that every gate other than the output has only polylog n remaining children. Finally, as the last step, we use a recent PRG by Meka, Reingold, and Tal [32] to fool this simpler formula.