Z-reachability Problem for Games on 2-dimensional Vector Addition Systems with States is in P

We consider a two-player infinite game with zero-reachability objectives played on a 2-dimensional vector addition system with states (VASS), the states of which are divided between the two players. Brazdil, Jancar, and Kucera (2010) have shown that for k > 0, deciding the winner in a game on k-dimensional VASS is in (k - 1)-EXPTIME. In this paper, we show that, for k = 2, the problem is in P, and thus improve the EXPTIME upper bound.