Reachability in Vector Addition System with States Parameterized by Geometric Dimension

TL;DR

This paper investigates the reachability problem in Vector Addition Systems with States parameterized by geometric dimension, revealing complexity results that differ from traditional approaches and extending existing techniques.

Contribution

It introduces the concept of geometric dimension for VASS and proves PSPACE-completeness of reachability for 1D and 2D cases using extended pumping techniques.

Findings

Reachability in 1D and 2D VASS is PSPACE-complete.

Geometric dimension offers a new perspective on VASS analysis.

Extended pumping techniques facilitate complexity proofs.

Abstract

The geometric dimension of a Vector Addition System with States (VASS), emerged in Leroux and Schmitz (2019) and formalized by Fu, Yang, and Zheng (2024), quantifies the dimension of the vector space spanned by cycle effects in the system. This paper explores the VASS reachability problem through the lens of geometric dimension, revealing key differences from the traditional dimensional parameterization. Notably, we establish that the reachability problem for both geometrically 1-dimensional and 2-dimensional VASS is PSPACE-complete, achieved by extending the pumping technique originally proposed by Czerwi\'nski et al. (2019).