The Complexity of Reachability in Affine Vector Addition Systems with States

TL;DR

This paper classifies the computational complexity of reachability problems in affine vector addition systems with states, revealing a detailed landscape that varies from NP-complete to undecidable depending on the affine operations involved.

Contribution

It provides a comprehensive complexity classification (trichotomy and dichotomy) for reachability in affine VASS, extending understanding beyond standard VASS.

Findings

NP-complete for VASS with resets

PSPACE-complete for VASS with transfers and copies

Undecidable for other affine classes

Abstract

Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations, e.g., allowing for negative intermediate counter values. It is natural to question their feasibility for VASS enriched with primitives that typically translate into undecidability. Spurred by this concern, we pinpoint the complexity of integer relaxations with respect to arbitrary classes of affine operations. More specifically, we provide a trichotomy on the complexity of integer reachability in VASS extended with affine operations (affine VASS). Namely, we show that it is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. We further present a dichotomy for standard reachability in affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. This yields a complete and unified complexity landscape of reachability in affine VASS. We also consider the reachability problem parameterized by a fixed affine VASS, rather than a class, and we show that the complexity landscape is arbitrary in this setting.