# 22.3 Universal coverability for Orthant VAS

An orthant is a region in $$\mathbb{R}^d$$ in which every vector in the orthant has the same sign in each dimension. In a vector addition system sum vectors taken non-deterministically from a given set, providing the sum remains in the positive orthant (all vectors positive). Z-VAS can visit all orthants, but the set of vectors that can be applied is the same in every orthant.

An orthant VAS has a different set of vectors for each orthant. Any vector from the set of the current orthant can be applied, the result of which may be in the same orthant, or another orthant.

The sets of vectors are monotonically increasing; so that if a vector can be applied in some orthant, it can also be applied in every orthant with additional positive dimensions.

The coverability problem asks whether the positive orthant can be reached from a given point and is undecidable. Thus too is reachability.

The universal coverability problem asks whether the positive orthant can be reached from every starting point. Is known to be decidable in dimension 3.

Question: can this be extended to dimension 4+. Is it also undecidable in general?