# 19.10 Variants of one-counter systems universality

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?