to $q_6$.

Bounded Universality {#sec:NondetBounded}
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We show that the problem is undecidable by making some changes to the undecidability proof of the initial-value universality problem.

Given a two-counter machine $\M$, we construct a one-counter net $\A'$ that is similar to $\A$, as constructed above, except for the following changes (see Figure \[fig:A’\]):

-   There is an additional state $q'_0$ that is accepting, it is the new initial state, and it has a nondeterministic choice over $\Sigma$ of either taking a self loop with weight $+1$ or going to $q_0$ with weight $0$.

-   The state $q_0$ is no longer initial, and it has an additional transition over $\#$ to a new state $q_7$ with weight $0$.

-   The state $q_7$ is accepting, and it has nondeterministic choice over $\Sigma$ of either taking a self loop with weight $-1$ or going to $q_0$ with weight $-1$.

Now $\M$ halts if and only if $\A'$ is bounded universal for an initial counter value $0$. A detailed proof can be found in \[apx:thm:Undecidable1\].

[theorem]{}[undecBU]{} \[thm:Undecidable2\] The bounded universality problem for one-counter nets is undecidable.

Singleton Alphabet {#sec:unary}
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In this section we study universality problems on OCN over singleton alphabets. The universality 