  One-Counter Nets (OCNs) are finite-state automata equipped with a counter
that is not allowed to become negative, but does not have zero tests. Their
simplicity and close connection to various other models (e.g., VASS, Counter
Machines and Pushdown Automata) make them an attractive model for studying the
border of decidability for the classical decision problems.
  The deterministic fragment of OCNs (DOCNs) typically admits more tractable
decision problems, and while these problems and the expressive power of DOCNs
have been studied, the determinization problem, namely deciding whether an OCN
admits an equivalent DOCN, has not received attention.
  We introduce four notions of OCN determinizability, which arise naturally due
to intricacies in the model, and specifically, the interpretation of the
initial counter value. We show that in general, determinizability is
undecidable under most notions, but over a singleton alphabet (i.e., 1
dimensional VASS) one definition becomes decidable, and the rest become
trivial, in that there is always an equivalent DOCN.

Formal Languages and Automata Theory