Regular separability of one counter automata

The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSPACE-completeness, of the regular separability problem for languages of one counter automata without zero tests (also known as one counter nets). This contrasts with undecidability of the regularity problem for one counter nets, and with undecidability of the regular separability problem for one counter automata, which is our second result.