Weak Lower Bounds on Resource-Bounded Compression Imply Strong Separations of Complexity Classes

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.