The complexity of regular abstractions of one-counter languages

We study the computational and descriptional complexity of the following transformation: Given a one-counter automaton (OCA) $\mathcal{A}$, construct a nondeterministic finite automaton (NFA) $\mathcal{B}$ that recognizes an abstraction of the language $\mathcal{L}\left( \mathcal{A} \right)$: its (1) downward closure, (2) upward closure, or (3) Parikh image. For the Parikh image over a fixed alphabet and for the upward and downward closures, we find polynomial-time algorithms that compute such an NFA. For the Parikh image with the alphabet as part of the input, we find a quasi-polynomial time algorithm and prove a completeness result: we construct a sequence of OCA that admits a polynomial-time algorithm iff there is one for all OCA. For all three abstractions, it was previously unknown whether appropriate NFA of sub-exponential size exist.