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

Abstract

For a size parameter \(s:\mathbb {N}\to \mathbb {N}\), 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}ⁿ →{0,1} (represented by a string of length N := 2ⁿ) 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 \(\text {MCSP}[2^{\mu _{1}\cdot n}]\) cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^1.01, then P≠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 \(\text {MCSP}[2^{\mu _{2}\cdot n}]\) in time N^1.99, for some constant μ₂ > μ₁. (2) A non-deterministic (or parity) branching program of size \(o(N^{1.5}/\log N)\) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nečiporuk method to MKTP, which previously appeared to be difficult. (3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least \(N^{1.5-o\left (1\right )}\). 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, Electron. Colloq. Comput. Complexity (ECCC) 26, 51, 2019). En route, we obtain several related results: (1) There exists a (local) hitting set generator with seed length \(\widetilde {O}(\sqrt {N})\) 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 \(2^{\widetilde {\Omega }(N)}\).