High-probability version under accessibility of the added transition’s source

Determine whether, in the uniform random model of almost deterministic n-state automata (a random deterministic complete transition structure on a fixed finite alphabet with one additional random transition p→a→q) and a uniformly random initial state, conditioning that the source state p of the added transition is accessible from the initial state implies that the language’s minimal deterministic automaton has super-polynomial size with high probability as n→∞ (i.e., the probability tends to 1 that the minimal DFA has more than n^d states for every fixed d≥1).

Background

The paper proves that for the random model of almost deterministic automata, the state complexity of the recognized language is super-polynomial with visible probability (a constant bounded away from 0). This cannot hold with high probability in the unconstrained model because, with asymptotically constant probability, the source of the added transition lies outside the accessible part and thus the added transition does not affect the language.

The authors suggest that conditioning the source of the added transition to be accessible could plausibly yield a high-probability version of their super-polynomial lower bound, but their current techniques (based on Birthday Problem–style arguments) are insufficient to establish such a result.