The reachability problem for two-dimensional vector addition systems with states

We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that if a configuration is reachable then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This in turn enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.