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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

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).

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)

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@InProceedings{limaye_et_al:LIPIcs.FSTTCS.2019.22, author = {Limaye, Nutan and Srinivasan, Srikanth and Tripathi, Utkarsh}, title = {{More on AC^0\lbrackoplus\rbrack and Variants of the Majority Function}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {22:1--22:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.22}, URN = {urn:nbn:de:0030-drops-115844}, doi = {10.4230/LIPIcs.FSTTCS.2019.22}, annote = {Keywords: AC^0\lbrackoplus\rbrack, Coin Problem, Promise Majority} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

The probabilistic degree of a Boolean function f:{0,1}^n -> {0,1} is defined to be the smallest d such that there is a random polynomial P of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions - specifically symmetric Boolean functions - have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems.
In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. On the Probabilistic Degrees of Symmetric Boolean Functions. 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. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{srinivasan_et_al:LIPIcs.FSTTCS.2019.28, author = {Srinivasan, Srikanth and Tripathi, Utkarsh and Venkitesh, S.}, title = {{On the Probabilistic Degrees of Symmetric Boolean Functions}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {28:1--28:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.28}, URN = {urn:nbn:de:0030-drops-115908}, doi = {10.4230/LIPIcs.FSTTCS.2019.28}, annote = {Keywords: Symmetric Boolean function, probabilistic degree} }

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