Starting from finite state automata, one can create a hierarchy of automata models by repeatedly applying one of the following constructions:

- Adding VASS counters (i.e. over \(\mathbb{N}\), without zero tests), and
- Building stacks containing the model.

As a special case, we get, for example:

- \(\mathsf{FSA}\) with \(\mathbb{N}\)-counters \(= \mathsf{VASS}\)
- Stacks over \(\mathsf{FSA}\) \(= \mathsf{PDA}\)
- \(\mathsf{PDA}\) with \(\mathbb{N}\)-counters \(= \mathsf{PVASS}\)

It would be interesting to know whether this hierarchy is proper, or if it collapses at some level.

A first avenue of investigation might be to consider the hierarchy created by building stacks and adding \(\mathbb{Z}\)-counters.