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# More on AC^0[oplus] and Variants of the Majority Function

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## Cite As

Nutan Limaye, Srikanth Srinivasan, and Utkarsh Tripathi. More on AC^0[oplus] and Variants of the Majority Function. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.FSTTCS.2019.22

## Abstract

In this paper we prove two results about AC^0[oplus] circuits. (1) We show that for d(N) = o(sqrt(log N/log log N)) and N <= s(N) <= 2^(dN^(1/4d^2)) there is an explicit family of functions {f_N:{0,1}^N - > {0,1}} such that - f_N has uniform AC^0 formulas of depth d and size at most s; - f_N does not have AC^0[oplus] formulas of depth d and size s^epsilon, where epsilon is a fixed absolute constant. This gives a quantitative improvement on the recent result of Limaye, Srinivasan, Sreenivasaiah, Tripathi, and Venkitesh, (STOC, 2019), which proved a similar Fixed-Depth Size-Hierarchy theorem but for d << log log N and s << exp(N^(1/2^Omega(d))). As in the previous result, we use the Coin Problem to prove our hierarchy theorem. Our main technical result is the construction of uniform size-optimal formulas for solving the coin problem with improved sample complexity (1/delta)^O(d) (down from (1/delta)^(2^O(d)) in the previous result). (2) In our second result, we show that randomness buys depth in the AC^0[oplus] setting. Formally, we show that for any fixed constant d >= 2, there is a family of Boolean functions that has polynomial-sized randomized uniform AC^0 circuits of depth d but no polynomial-sized (deterministic) AC^0[oplus] circuits of depth d. Previously Viola (Computational Complexity, 2014) showed that an increase in depth (by at least 2) is essential to avoid superpolynomial blow-up while derandomizing randomized AC^0 circuits. We show that an increase in depth (by at least 1) is essential even for AC^0[oplus]. As in Viola’s result, the separating examples are promise variants of the Majority function on N inputs that accept inputs of weight at least N/2 + N/(log N)^(d-1) and reject inputs of weight at most N/2 - N/(log N)^(d-1).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Circuit complexity
• Theory of computation → Pseudorandomness and derandomization
##### Keywords
• AC^0[oplus]
• Coin Problem
• Promise Majority

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