Functions Weakly-Computable by Vector Addition Systems

A vector addition system with states (VASS) is a finite graph $(Q,E)$ with labels in $\mathbb{Z}^d$ and a designated initial state $q_0$ and final state $q_f$. The set of configurations is $Q \times \mathbb{N}^d$, and a step according to an edge $e=(p,q)$ labelled with $v$ consists of moving from the current configuration $p,x$ to $q, x+v$, where $x+v$ has to be in $\mathbb{N}^d$ again. The standard definition of computing a function with a VASS is as follows: One of the $d$ "counters", usually the first, is a designated input counter. Another counter, usually the second, is used for the output. The rest of the counters are called auxiliary counters. A VASS weakly-computes a function f if the following two conditions hold: For all $x \in \mathbb{N}$, there exists some vector $w' \in \mathbb{N}^{d-2}$ and $x' \in \mathbb{N}$ such that $q_0, (x,0,0^{d-2})$ can reach $q_f, (x', f(x), w')$. I.e. the VASS can reach the final state with the value $f(x)$ in the output counter, the values left in auxiliary counters are irrelevant. For all $x,x',y' \in \mathbb{N}$ and vectors $w' \in \mathbb{N}^{d-2}$, if $q_0, (x,0,0^{d-2})$ can reach $q_f, (x', y', w')$, then $y' \leq f(x)$. I.e. if the VASS can reach the final state with a value $y'$ in the output counter, again irrespective of values left in the auxiliary counters, then $y' \leq f(x)$. Despite this being the standard definition, surprisingly little is known about functions weakly-computable by vector addition systems. Essentially the only known properties are the following. They are monotone, i.e. if $x \leq x'$, then $f(x) \leq f(x')$. They are primitive-recursive, in particular computable. They grow at least linearly, or intuitively "have to grow steeper over time". To resolve the current state of no progress in this direction, I suggest the following open problems: Is $2^{\sqrt{x}}$ weakly-computable? It would be quite surprising, since $\sqrt{x}$ is not. Which linear recursive sequences (LRS) are weakly-computable? I conjecture that exactly the monotone LRS are weakly computable (asymptotically at least).