# Petri nets: Rackoff coverage limits in case the VAS addition vectors and the target vector are {-1, 0, +1}?

In "Coverage problems and limits for vector addition systems," Rackoff considers a VAS $$(v, A)$$ of dimension $$k$$ and size $$n$$ and derives an upper limit of $$2 ^ 2 ( log_2 3) n ( log_2 n)}}$$ on the duration of non-negative coverage executions.

Consider the case. $$A subseteq {- 1.0, + 1 } ^ k$$, $$v in mathbb {N} ^ k$$ and the vector to cover be of $${0,1 } ^ k$$.

What would be a good upper limit on the duration of covering non-negative executions in terms of $$k$$? Using Rackoff & # 39; s Thm. 3.5, $$lvert A rvert leqslant 3 ^ k$$ and $$n = mathcal {O} (3 ^ k + | v | _1)$$ (where $$| cdot | 1$$ returns rule 1 of a vector) would produce an upper limit of $$2 ^ {2 ^ { mathcal {O} (3 ^ k + | v | _1) log_2 (3 ^ k + | v | _1)}}$$. We will try to make it more strict in this particular case.

It seems to me that a better limit would result from a better limit in $$f (k)$$ with respect to $$k$$ (instead of $$n$$), where
$$f (0) = 1$$
and
$$f (i + 1) leqslant (2 ^ n f (i)) ^ {i + 1} + f (i) qquad text {for i
Any idea how to tie $$f (k)$$ for an expression in $$k$$? If I interpret the Thm 3.5 test correctly, we could probably reach $$2 ^ (3k) ^ k}$$ as an upper limit on the length of covering non-negative executions. Can you confirm or reject this? Is a tighter limit possible in the case considered?