alphabet. Due to their lack of an explicit zero-test, OCNs are monotone with respect to counter values: if it is possible to make an $a$-labelled step from a configuration with state $p$ and counter $n$ to state $q$ with counter $n+d$, written as $(p,n) \step{a} (q,n+d)$ here, then the same holds for any larger counter value $m\ge n$: $(p,m) \step{a}(q,m+d)$. Consequently, if we define the language via acceptance by reaching a final control state, then for all states $s$ and $n\le m\in\N$, the language $\Lang{s,n}$ of the initial configuration $(s,n)$ is included in that of $(s,m)$. This motivates our first variation of the universality problem. The *Initial-Value Universality* problem asks if there exists a sufficiently large initial counter to make the resulting language universal.

The second question we consider is the *Bounded Universality* problem, which asks if there exists a large enough upper bound on the counter so that every word can be accepted via a run that remains within this bound. Writing $\bLang{\initialstate,c_0}{b}\subseteq \alphabet^*$ for the $b$-bounded language from configuration $(\initialstate,c_0)$, the decision problem is as follows.

The motivation for studying these parameterized problems comes from the observation that the “vanilla” universality problem, without existentially quantifying over parameters, is 