One-way alternating pushdown automata (1APDA) can recognize any language in $ DTIME(2^{O(n)}) $ (Alternation by Chandra, Kozen, and Stockmeyer, 1981). By replacing a pushdown storage of a 1APDA with a counter, we can obtain a one-way alternating automaton with one-counter (1ACA). My question is about 1ACAs on unary languages.
Can 1ACAs recognize some unary non-regular languages?
Note that one-way nondeterministic pushdown automata can recognize only unary regular languages.
Yes. Consider the language $L = \{ a^n \mid n = 2^s, s \geq 0 \}$ and construct a one-way alternating one-counter automaton recognizing $L$ in the following way. First, the automaton starts increasing the value of the counter and guesses when to stop, that is, guesses some value $m$. Then it branches universally: the first branch checks that the length of the input is precisely $2 m$, and the second branch moves $m$ cells forward on the input and checks that the remainder is in $L$, by going to the initial control state. Now add a base case: let the device accept if the input tape is exactly of length $1$, by making a nondeterministic guess at the initial state. That completes the construction.
In a similar fashion, one can get products of the form $n = k_1^{s_1} \ldots k_r^{s_r}$, with $k_1, \ldots, k_r$ fixed and $s_1, \ldots, s_r$ arbitrary.
