AC0 Unpredictability

Abstract

We prove that for every distribution D on n bits with Shannon entropy ≥ n − a, at most O(2da logd+1g)/γ5 of the bits Di can be predicted with advantage γ by an AC0 circuit of size g and depth D that is a function of all of the bits of D except Di. This answers a question by Meir and Wigderson, who proved a corresponding result for decision trees. We also show that there are distributions D with entropy ≥ n − O(1) such that any subset of O(n/ log n) bits of D on can be distinguished from uniform by a circuit of depth 2 and size poly(n). This separates the notions of predictability and distinguishability in this context.