Weak lower bounds on resource-bounded compression imply strong separations of complexity classes

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 using bounded-size circuits. 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[2√n], or lower bounds for approximating MCSP[2o(n)], would imply major separations such as NP ⊄BPP and NP ⊄P/poly. We strengthen their 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 Σ3 SAT oracle gates of small fan-in (related to the size s(n) we want to test). For A : {0,1}⋆ → {0,1}, let search-oracleMCSPA[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/log⋆ n 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 ⊄P/poly. (3) If search-MCSP[s(n)] does not have N · poly(s(n))-size, O(logN)-depth circuits, then NP ⊄NC1. Note it is known that MCSP[2√n] 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 ⊄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.