# Masaryk University

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# Efficient Analysis of VASS Termination Complexity

Authors LEROUX Jérôme 2020 Article in Proceedings LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science http://dx.doi.org/10.1145/3373718.3394751 Vector addition systems; Termination The termination complexity of a given VASS is a function $L$ assigning to every $n$ the length of the longest nonterminating computation initiated in a configuration with all counters bounded by $n$. We show that for every VASS with demonic nondeterminism and every fixed $k$, the problem whether $L \in G_k$, where $G_k$ is the $k$-th level in the Grzegorczyk hierarchy, is decidable in polynomial time. Furthermore, we show that if $L \notin G_k$, then L grows at least as fast as the generator $F_k+1$ of $G_k+1$. Hence, for every terminating VASS, the growth of $L$ can be reasonably characterized by the least $k$ such that $L \in G_k$. Furthermore, we consider VASS with both angelic and demonic nondeterminism, i.e., VASS games where the players aim at lowering/raising the termination time. We prove that for every fixed $k$, the problem whether $L \in G_k$ for a given VASS game is NP-complete. Furthermore, if $L \notin G_k$, then $L$ grows at least as fast as $F_k+1$.