Improved Pseudorandom Generators for 𝖠𝖒^0 Circuits

by   Xin Lyu, et al.

We show a new PRG construction fooling depth-d, size-m 𝖠𝖒^0 circuits within error Ξ΅, which has seed length O(log^d-1(m)log(m/Ξ΅)loglog(m)). Our PRG improves on previous work (Trevisan and Xue 2013, Servedio and Tan 2019, Kelley 2021) from various aspects. It has optimal dependence on 1/Ξ΅ and is only one β€œloglog(m)” away from the lower bound barrier. For the case of d=2, the seed length tightly matches the best-known PRG for CNFs (De et al. 2010, Tal 2017). There are two technical ingredients behind our new result; both of them might be of independent interest. First, we use a partitioning-based approach to construct PRGs based on restriction lemmas for 𝖠𝖒^0, which follows and extends the seminal work of (Ajtai and Wigderson 1989). Second, improving and extending prior works (Trevisan and Xue 2013, Servedio and Tan 2019, Kelley 2021), we prove a full derandomization of the powerful multi-switching lemma for a family of DNFs (HΓ₯stad 2014).


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