Reachability in Two-Dimensional Vector Addition Systems with States: One Test is for Free

TL;DR

This paper proves that the reachability problem in a two-dimensional vector addition system with states extended with zero tests remains solvable in polynomial space, extending previous results to a more powerful model.

Contribution

It introduces the 2-TVASS model with zero tests and shows that reachability remains polynomial space solvable, expanding the understanding of computational complexity in this domain.

Findings

Reachability in 2-TVASS is in PSPACE.

Small reachability certificates exist for 2-TVASS.

Extension of classical models with zero tests still allows polynomial space solutions.

Abstract

Vector addition system with states is an ubiquitous model of computation with extensive applications in computer science. The reachability problem for vector addition systems is central since many other problems reduce to that question. The problem is decidable and it was recently proved that the dimension of the vector addition system is an important parameter of the complexity. In fixed dimensions larger than two, the complexity is not known (with huge complexity gaps). In dimension two, the reachability problem was shown to be PSPACE-complete by Blondin et al. in 2015. We consider an extension of this model, called 2-TVASS, where the first counter can be tested for zero. This model naturally extends the classical model of one counter automata (OCA). We show that reachability is still solvable in polynomial space for 2-TVASS. As in the work Blondin et al., our approach relies on the existence of small reachability certificates obtained by concatenating polynomially many cycles.