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One-counter nets (OCN) are Petri nets with exactly one unbounded place. They are equivalent to a subclass of one-counter automata with only a weak test for zero.
We show that weak simulation preorder is decidable for OCN and that weak simulation approximants do not converge at level \(\omega\), but only at \(\omega^2\). In contrast, other semantic relations like weak bisimulation are undecidable for OCN, and so are weak (and strong) trace inclusion.