When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2019.66
URN: urn:nbn:de:0030-drops-106422
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10642/
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### AC^0[p] Lower Bounds Against MCSP via the Coin Problem

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### Abstract

Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean function has circuit complexity less than a given parameter s. We prove that MCSP is hard for constant-depth circuits with mod p gates, for any prime p >= 2 (the circuit class AC^0[p]). Namely, we show that MCSP requires d-depth AC^0[p] circuits of size at least exp(N^{0.49/d}), where N=2^n is the size of an input truth table of an n-variate boolean function. Our circuit lower bound proof shows that MCSP can solve the coin problem: distinguish uniformly random N-bit strings from those generated using independent samples from a biased random coin which is 1 with probability 1/2+N^{-0.49}, and 0 otherwise. Solving the coin problem with such parameters is known to require exponentially large AC^0[p] circuits. Moreover, this also implies that MAJORITY is computable by a non-uniform AC^0 circuit of polynomial size that also has MCSP-oracle gates. The latter has a few other consequences for the complexity of MCSP, e.g., we get that any boolean function in NC^1 (i.e., computable by a polynomial-size formula) can also be computed by a non-uniform polynomial-size AC^0 circuit with MCSP-oracle gates.

### BibTeX - Entry

```@InProceedings{golovnev_et_al:LIPIcs:2019:10642,
author =	{Alexander Golovnev and Rahul Ilango and Russell Impagliazzo and Valentine Kabanets and Antonina Kolokolova and Avishay Tal},
title =	{{AC^0[p] Lower Bounds Against MCSP via the Coin Problem}},
booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
pages =	{66:1--66:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-109-2},
ISSN =	{1868-8969},
year =	{2019},
volume =	{132},
editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},