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}^n -> {0,1} (represented by a string of length N := 2^n)
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 \mu_1 > 0, if
MCSP[2^{\mu_1 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 MCSP[2^{\mu_2 n}] in time N^{1.99},
for some constant \mu_2 > \mu_1.
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 Neciporuk 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(1)}.
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
\tilde{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^{\tilde{\Omega}(N)}.
(To appear
STACS 2021).
