Problem 1:^{1} Given a 1-VASS, let \(L_n\) be its language where
acceptance is by reaching a final state from a fixed initial state and initial
counter value \(n\). Does there exist \(n\) such that \(\Sigma^* = L_n\) ?

Problem 2: Given a 1-VASS, let \(L^n\) be the language of the $n$-bounded system (the NFA where values 0..n are hard-coded) where acceptance is by reaching a final state from a fixed initial configuration. Does there exist \(n\) such that \(\Sigma^* \subseteq L^n\)?

Questions:

- Are these problems decidable?
- Are they inter-reducible?
- What about trace languages?
- Is there a direct reduction from the seemingly simpler Universality Problem (fixed initial configuration) to either of these problems?

## Footnotes:

^{1}

Whether this problem is decidable is also a question suggested by Piotrek Hofman