License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10642/
Golovnev, Alexander ;
Ilango, Rahul ;
Impagliazzo, Russell ;
Kabanets, Valentine ;
Kolokolova, Antonina ;
Tal, Avishay
AC^0[p] Lower Bounds Against MCSP via the Coin Problem
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},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10642},
URN = {urn:nbn:de:0030-drops-106422},
doi = {10.4230/LIPIcs.ICALP.2019.66},
annote = {Keywords: Minimum Circuit Size Problem (MCSP), circuit lower bounds, AC0[p], coin problem, hybrid argument, MKTP, biased random boolean functions}
}
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Keywords: |
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Minimum Circuit Size Problem (MCSP), circuit lower bounds, AC0[p], coin problem, hybrid argument, MKTP, biased random boolean functions |
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Collection: |
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46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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04.07.2019 |
