Relations and equivalences between circuit lower bounds and Karp-Lipton theorems
Name
LIPIcs-CCC-2019-30.pdf
Description
Published version
Size
752.95 KB
Format
Adobe PDF
Checksum (MD5)
fec61a64d954c1b8f69bb46cdd9ef05e
Author(s)
Williams, Richard Ryan
Murray, Cody
Chen, Lijie
McKay, Dylan M.
Date Issued
2019
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
Williams, Richard Ryan, Murray, Cody, Chen, Lijie and McKay, Dylan M. 2019. "Relations and equivalences between circuit lower bounds and Karp-Lipton theorems." Leibniz International Proceedings in Informatics, LIPIcs, 137.
Version
Final published version
Abstract
© Lijie Chen, Dylan M. McKay, Cody D. Murray, and R. Ryan Williams; licensed under Creative Commons License CC-BY 34th Computational Complexity Conference (CCC 2019). A frontier open problem in circuit complexity is to prove PNP 6⊂ SIZE[nk] for all k; this is a necessary intermediate step towards NP 6⊂ P/poly. Previously, for several classes containing PNP, including NPNP, ZPPNP, and S2P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove C 6⊂ SIZE[nk] for all k, we show that C ⊂ P/poly implies a “collapse” D = C for some larger class D, where we already know D 6⊂ SIZE[nk] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for PNP 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 PNP are equivalent to fixed-polynomial size circuit lower bounds for PNP. That is, PNP 6⊂ SIZE[nk] for all k if and only if (NP ⊂ P/poly implies PH ⊂ i.o.-PNP/n). Next, we present new consequences of the assumption NP ⊂ 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 6⊂ SIZE[nk] for all k, then for all C ∈ {P, PP, PSPACE, EXP}, C ⊂ P/poly implies C ⊂ i.o.-NP/nε for all ε > 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 n1+ε-size circuits, then MA ⊂ i.o.-NP/O(log n), MA ⊂ i.o.-PNP[O(log n)], and NEXP 6⊂ SIZE[2o(m)]. Finally, we observe connections between these results and the “hardness magnification” phenomena described in recent works.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.CCC.2019.30
