One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

Abstract

For a size parameter s : N → N, the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0, 1} → {0, 1} (represented by a string of length N := 2) is at most a threshold s(n). A recent line of work exhibited “hardness magnification” phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant μ1 > 0, if MCSP[2μ1·n] cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N1.01, then P 6= NP. In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: 1. A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute MCSP[2μ2·n] in time N1.99, for some constant μ2 > μ1. 2. A non-deterministic (or parity) branching program of size o(N1.5/ logN) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nechiporuk method to MKTP, which previously appeared to be difficult. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models. The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola 2019). En route, we obtain several related results: 1. There exists a (local) hitting set generator with seed length Õ( √ N) secure against read-once polynomial-size non-deterministic branching programs on N -bit inputs. 2. Any read-once co-non-deterministic branching program computing MCSP must have size at least 2Ω̃(N). 2012 ACM Subject Classification Theory of computation → Circuit complexity; Theory of computation → Pseudorandomness and derandomization