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

For a size parameter s: ? ? ?, 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 ?? > 0, if MCSP[2^{??? 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^{???n}] in time N^{1.99}, for some constant ?? > ??. 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 Ne?iporuk 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 O?(?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^??(N)

BEARALERTS: A successful flare prediction system

We describe our BEARALERT program of predicting solar flares or rapid development of activity in certain sunspot groups. The purpose of the program is to test our understanding of the flare process by making public predictions via electronic mail. Neither the exact timing of the flare nor the possibility of emergence of new active regions can be predicted. But high-resolution observations of the magnetic configuration, Ha brightness and structure and other properties of a region enabled us to announce the onset of 15 of 23 major active regions over a two-year period, and 15 of 32 BEARALERTS were followed by this activity. We used high-resolution real-time data available at the Big Bear Solar Observatory (BBSO). The criteria for prediction are given and discussed, along with those for filament eruption. The success fo the BEARALERT is evaluated by counting the M- and X-class flares in six days following the alert and comparing these results with those of a number of other predictive schemes. We find the single regions chosen had about 30% more flares than the whole disk on random days, or several times more than individual regions chosen at random. There was a gain of 1.5 to 2.0 times in flare frequency compared to regions selected by spot size or complexity. We also find an improvement of 20–40% over large or complex regions that have had some flares already. The ratio of improvement has increased with time as we gained experience. In the 24-hr period following each alert, one or more M-class or greater flares occurred 72% of the time. We also checked the possibility of prediction by the 152-day interval which some workers have claimed, but found those results slightly worse than random and considerably inferior to the BEARALERTS. All of the particularly active regions that were missed either occurred during bad weather at BBSO or were missed because we only issued alerts for one region at a time