We consider the model of history-deterministic one-counter nets (OCNs). History-determinism is a property of transition systems that allows for a limited kind of non-determinism which can be resolved ‘on-the-fly’. Token games, which have been used to characterise history-determinism over various models, also characterise history-determinism over OCNs. By reducing 1-token games to simulation games, we are able to show that checking for history-determinism of OCNs is decidable. Moreover, we prove that this problem is $$\textbf{PSPACE}$$ PSPACE -complete for a unary encoding of transitions, and $$\textbf{EXPSPACE}$$ EXPSPACE -complete for a binary encoding and undecidable for one-counter automata (OCA), which are OCNs that can test for zeroes.We then study the language properties of history-deterministic OCNs. We show that the resolvers of non-determinism for history-deterministic OCNs are eventually periodic. As a consequence, for a given history-deterministic OCN, we construct a language equivalent deterministic OCA. We also show the decidability of comparing languages of history-deterministic OCNs, such as language inclusion and language universality.
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