# On Functions Weakly Computable by Pushdown Petri Nets and Related Systems

## TL;DR

This paper investigates the computational capabilities of grammar-controlled vector addition systems (GVASes), showing they can compute certain fast-growing functions but not their inverses or sublinear functions, with a new pumping lemma proof.

## Contribution

It introduces a pumping lemma for GVAS runs and characterizes the class of functions weakly computable by GVASes, highlighting their limitations and strengths.

## Key findings

- GVASes can weakly compute all functions $F_\alpha$ for $\alpha<\omega^\omega$
- GVASes cannot weakly compute inverse functions $F_\alpha^{-1}$ or sublinear functions
- A new pumping lemma for GVAS runs is established

## Abstract

We consider numerical functions weakly computable by grammar-controlled vector addition systems (GVASes, a variant of pushdown Petri nets). GVASes can weakly compute all fast growing functions $F_\alpha$ for $\alpha<\omega^\omega$, hence they are computationally more powerful than standard vector addition systems. On the other hand they cannot weakly compute the inverses $F_\alpha^{-1}$ or indeed any sublinear function. The proof relies on a pumping lemma for runs of GVASes that is of independent interest.