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

$$ f (i + 1) leqslant (2 ^ n f (i)) ^ {i + 1} + f (i) qquad text {for $ i <k $} $$

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?