Exponential-size strengthening of the super-polynomial lower bound

Ascertain whether, for the uniform random model of almost deterministic n-state automata described in the paper, the language’s minimal deterministic automaton has exponential size in n (for example, of order c^n for some c>1), rather than merely super-polynomial, in the asymptotic regime n→∞.

Background

The main theorem shows that the state complexity is super-polynomial with visible probability. A natural question is whether a stronger exponential lower bound can be established in the same random model.

The authors indicate that current methods do not suffice to prove exponential growth, suggesting that new ideas beyond the paper’s techniques would be required.