## 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?

## The rules implicitly assume that non-two-handed weapons are one-handed.

The two-handed property says:

This weapon requires two hands when you attack with it.

For weapons without two-handed ownership, I cannot find anything in the rules that directly say that they are considered with only one hand. However, although there is no explicit ownership of "one-handed" weapons, there is strong circumstantial evidence that designers consider that all non-two-handed weapons are one-handed weapons. First, the feat of Crossbow Expert and the ownership of ammunition, both make explicit reference to "one-handed weapons." In addition, the rules for melee attacks say:

A melee attack usually uses a hand weapon such as a sword, a war hammer or an ax.

Similarly, the rules for improvised weapons also refer to holding the weapon in your hands:

An improvised weapon includes any object you can wield with one or two hands, such as broken glass, a table leg, a pan, a wagon wheel or a dead goblin.

And assuming that all PCs have 2 hands (which is currently the case for all official playable races), the only ways to wield a handgun are with 1 hand or 2.

Finally, some official decisions in the Wise Counseling Compendium also refer to one-handed weapons, such as:

### Double carrier

Can my character wield two heavy weapons if he has the Dual Wielder feat? No. The Dual Wielder feat allows you to handle two weapons with one hand. Each heavy weapon in the Player Manual also has the property of two hands.

This decision strongly implies that any weapon that is not two-handed is considered a one-handed weapon, or else would not be eligible for the fight with two weapons. In general, although it is never explicitly stated, the intention seems to be that each weapon requires 1 or 2 hands to handle it, so Any handgun that is not two-handed must be one-handed.