Consider the following universe problem.
The universe problem. Given a finite set $\Sigma$ for a class of languages, and an automaton accepting the language $L$, decide if $L=\Sigma^*$.
In [1], it is stated and proved that the universe problem is undecidable for a particular class of one-counter automata. This result then follows for the class of all non-deterministic one-counter automata. I'm wondering if it is known whether this problem is still undecidable when we restrict the size of the input alphabet of the automaton.
I think that with alphabet size 1 the problem becomes decidable, but what about size 2? And if that turns out to be decidable what is the smallest value of $n \in \mathbb{N}$ such that the problem is undecidable.
I think it's probable that the answer to this question is known but I'm having trouble finding an answer. If it is already known then I would appreciate a reference.