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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2019.30
URN: urn:nbn:de:0030-drops-108525
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10852/
Chen, Lijie ;
McKay, Dylan M. ;
Murray, Cody D. ;
Williams, R. Ryan
Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems
Abstract
A frontier open problem in circuit complexity is to prove P^{NP} is not in SIZE[n^k] for all k; this is a necessary intermediate step towards NP is not in P_{/poly}. Previously, for several classes containing P^{NP}, including NP^{NP}, ZPP^{NP}, and S_2 P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove C is not in SIZE[n^k] for all k, we show that C subset P_{/poly} implies a "collapse" D = C for some larger class D, where we already know D is not in SIZE[n^k] for all k.
It seems obvious that one could take a different approach to prove circuit lower bounds for P^{NP} that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for P^{NP} are equivalent to fixed-polynomial size circuit lower bounds for P^{NP}. That is, P^{NP} is not in SIZE[n^k] for all k if and only if (NP subset P_{/poly} implies PH subset i.o.- P^{NP}_{/n}).
Next, we present new consequences of the assumption NP subset P_{/poly}, towards proving similar results for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. That is, if NP is not in SIZE[n^k] for all k, then for all C in {Parity-P, PP, PSPACE, EXP}, C subset P_{/poly} implies C subset i.o.-NP_{/n^epsilon} for all epsilon > 0. Note that unconditionally, the collapses are only to MA and not NP.
We also explore consequences of circuit lower bounds for a sparse language in NP. Among other results, we show if a polynomially-sparse NP language does not have n^{1+epsilon}-size circuits, then MA subset i.o.-NP_{/O(log n)}, MA subset i.o.-P^{NP[O(log n)]}, and NEXP is not in SIZE[2^{o(m)}]. Finally, we observe connections between these results and the "hardness magnification" phenomena described in recent works.
BibTeX - Entry
@InProceedings{chen_et_al:LIPIcs:2019:10852,
author = {Lijie Chen and Dylan M. McKay and Cody D. Murray and R. Ryan Williams},
title = {{Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems}},
booktitle = {34th Computational Complexity Conference (CCC 2019)},
pages = {30:1--30:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-116-0},
ISSN = {1868-8969},
year = {2019},
volume = {137},
editor = {Amir Shpilka},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10852},
URN = {urn:nbn:de:0030-drops-108525},
doi = {10.4230/LIPIcs.CCC.2019.30},
annote = {Keywords: Karp-Lipton Theorems, Circuit Lower Bounds, Derandomization, Hardness Magnification}
}
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Keywords: |
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Karp-Lipton Theorems, Circuit Lower Bounds, Derandomization, Hardness Magnification |
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Collection: |
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34th Computational Complexity Conference (CCC 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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16.07.2019 |
