Reachability in Two-Dimensional Vector Addition Systems with States is   PSPACE-complete
  Determining the complexity of the reachability problem for vector addition systems with states (VASS) is a long-standing open problem in computer science. Long known to be decidable, the problem to this day lacks any complexity upper bound whatsoever. In this paper, reachability for two-dimensional VASS is shown PSPACE-complete. This improves on a previously known doubly exponential time bound established by Howell, Rosier, Huynh and Yen in 1986. The coverability and boundedness problems are also noted to be PSPACE-complete. In addition, some complexity results are given for the reachability problem in two-dimensional VASS and in integer VASS when numbers are encoded in unary. 
Michael Blondin, Alain Finkel, Stefan G\"oller, Christoph Haase,   Pierre McKenzie
cs.FL cs.CC cs.LO