Weak Lower Bounds on Resource-Bounded Compression Imply Strong Separations of Complexity Classes
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MCSP-MKTP-stoc19.pdf
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Accepted version
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Author(s)
McKay, Dylan M.
Murray, Cody D.
Williams, R. Ryan
Date Issued
June 2019
Journal
Proceedings of the Annual ACM Symposium on Theory of Computing
Publisher
Association for Computing Machinery (ACM)
Citation
McKay, Dylan M., Murray, Cody D. and Williams, R. Ryan. 2019. "Weak Lower Bounds on Resource-Bounded Compression Imply Strong Separations of Complexity Classes." Proceedings of the Annual ACM Symposium on Theory of Computing.
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Author's final manuscript
Abstract
© 2019 Copyright held by the owner/author(s). Publication rights licensed to ACM. The Minimum Circuit Size Problem (MCSP) asks to determine the minimum size of a circuit computing a given truth table. MCSP is a natural and powerful string compression problem whose NP-hardness remains open. Recently, Oliveira and Santhanam [FOCS 2018] and Oliveira, Pich, and Santhanam [ECCC 2018] demonstrated a “hardness magnification” phenomenon for MCSP in restricted settings. Letting MCSP[s(n)] be the problem of deciding if a truth table of length 2n has circuit complexity at most s(n), they proved that small (fixed-polynomial) average case circuit/formula lower bounds for MCSP[2n], or lower bounds for approximating MCSP[2o(n)], would imply major separations such as NP 1 BPP and NP 1 P/poly. We strengthen these results in several directions, obtaining magnification results from worst-case lower bounds on exactly computing the search version of generalizations of MCSP[s(n)], which also extend to time-bounded Kolmogorov complexity. In particular, we show that search-MCSP[s(n)] (where we must output a s(n)size circuit when it exists) admits extremely efficient AC0 circuits and streaming algorithms using Σ3SAT oracle gates of small fan-in (related to the size s(n) we want to test). For A : (0, 1) → (0, 1), let search-MCSPA[s(n)] be the problem: Given a truth table T of size N = 2n, output a Boolean circuit for T of size at most s(n) with AND, OR, NOT, and A-oracle gates (or report that no such circuit exists). Some consequences of our results are: (1) For reasonable s(n) ≥ n and A ∈ PH, if search-MCSPA[s(n)] does not have a 1-pass deterministic poly(s(n))-space streaming algorithm with poly(s(n)) update time, then P, NP. For example, proving that it is impossible to synthesize SAT-oracle circuits of size 2n/logn with a streaming algorithm on truth tables of length N = 2n using Nε update time and Nε space on length-N inputs (for some ε > 0) would already separate P and NP. Note that some extremely simple functions, such as EQUALITY of two strings, already satisfy such lower bounds. (2) If search-MCSP[nc] lacks Õ(N)-size, Õ(1)-depth circuits for a c ≥ 1, then NP 1 P/poly. (3) If search-MCSP[s(n)] doesn’t have circuits of N · poly(s(n)) size andO(log N) depth, then NP 1 NC1. It is known that MCSP[2n] does not have formulas of N1.99 size [Hirahara and Santhanam, CCC 2017]. (4) If there is an ε > 0 such that for all c ≥ 1, search-MCSP[2n/c] does not have N1+ε-size O(1/ε)-depth ACC0 circuits, then NP 1 ACC0. Thus the amplification results of Allender and Koucký [JACM 2010] can extend to problems in NP and beyond. Furthermore, if we substitute P, PP, PSPACE, or EXP-complete problems for the oracle A, we obtain separations for those corresponding complexity classes instead of NP. Analogues of the above results hold for time-bounded Kolmogorov complexity as well.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://doi.org/10.1145/3313276.3316396
