Tight Correlation Bounds for Circuits Between AC0 and TC0
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We initiate the study of generalized AC0 circuits comprised of negations and arbitrary unbounded fan-in gates that only need to be constant over inputs of Hamming weight ≥ k, which we denote GC0(k). The gate set of this class includes biased LTFs like the k-OR (output 1 iff ≥ k bits are 1) and k-AND (output 0 iff ≥ k bits are 0), and thus can be seen as an interpolation between AC0 and TC0. We establish a tight multi-switching lemma for GC0(k) circuits, which bounds the probability that several depth-2 GC0(k) circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-d size-s AC0 circuits lifts to depth-d size-s^.99 GC0(.01log s) circuits with no loss in parameters (other than hidden constants). Our result has the following applications: 1.Size-2^Ω(n^1/d) depth-d GC0(Ω(n^1/d)) circuits do not correlate with parity (extending a result of Håstad (SICOMP, 2014)). 2. Size-n^Ω(log n) GC0(Ω(log^2 n)) circuits with n^.249 arbitrary threshold gates or n^.499 arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). 3. There is a seed length O((log m)^d-1log(m/ε)loglog(m)) pseudorandom generator against size-m depth-d GC0(log m) circuits, matching the AC0 lower bound of Håstad stad up to a loglog m factor (extending a result of Lyu (CCC, 2022)). 4. Size-m GC0(log m) circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).
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