Scaling laws and time-to-reach estimates under boundary control

Determine precise scaling laws in the system size n for the reachability ratio ρ(n) and obtain accurate estimates of the minimal control time required to reach one configuration from another when controlling elementary cellular automata via boundary cells.

Background

Following the SAT-based experiments indicating a rapid decrease of the reachability ratio for many non–peripherally-linear ECA, the authors highlight the need for quantitative characterizations beyond qualitative trends. Specifically, they seek precise finite-size scaling laws for ρ(n) and estimates of the time required to steer one configuration to another under boundary control.

These questions relate to understanding how the structure of the configuration graph and the dynamics under boundary inputs evolve with n, and to quantifying the minimal time horizons needed for successful control when it is possible.