Reachability in VASS Extended with Integer Counters

TL;DR

This paper investigates the reachability problem in VASS extended with integer counters, establishing complexity bounds and demonstrating how integer counters affect the number of counters needed for hardness.

Contribution

It provides new complexity results for VASS+Z, including NP-completeness for 1-VASS+Z and upper bounds in higher dimensions, showing how integer counters influence computational complexity.

Findings

NP-complete reachability in 1-VASS+Z

Upper bounds in $ ext{F}_{d+2}$ for d-VASS+Z

Lower bounds showing reduced N-counter requirements due to Z-counters

Abstract

We consider a variant of VASS extended with integer counters, denoted VASS+Z. These are automata equipped with N and Z counters; the N-counters are required to remain nonnegative and the Z-counters do not have this restriction. We study the complexity of the reachability problem for VASS+Z when the number of N-counters is fixed. We show that reachability is NP-complete in 1-VASS+Z (i.e. when there is only one N-counter) regardless of unary or binary encoding. For $d \geq 2$, using a KLMST-based algorithm, we prove that reachability in d-VASS+Z lies in the complexity class $\mathcal{F}_{d+2}$. Our upper bound improves on the naively obtained Ackermannian complexity by simulating the Z-counters with N-counters. To complement our upper bounds, we show that extending VASS with integer counters significantly lowers the number of N-counters needed to exhibit hardness. We prove that reachability in unary 2-VASS+Z is PSPACE-hard; without Z-counters this lower bound is only known in dimension 5. We also prove that reachability in unary 3-VASS+Z is TOWER-hard. Without Z-counters, reachability in 3-VASS has elementary complexity and TOWER-hardness is only known in dimension 8.