Reachability in Vector Addition System with States Parameterized by Geometric Dimension
Yangluo Zheng
TL;DR
This paper introduces geometric dimension as a natural parameter for VASS reachability, defined as the dimension of the vector space spanned by cycle effects, and investigates how this parameter shapes complexity. It adapts a pumping-based technique to 2-dimensional geometrically-defined VASS, leveraging sign-reflecting and support projections to construct a canonical coordinate system within the 2D cycle space, enabling a PSPACE upper bound and matching hardness results. The main results show PSPACE-completeness for geometrically 2D and 1D VASS (binary encoding) and NP-completeness for geometrically 0D, while also providing a polynomial-time algorithm to compute the geometric dimension. The work highlights that geometric dimension offers a more faithful abstraction for reachability structure than traditional dimension, with implications for modeling interdependent system parameters and guiding future analyses beyond dimension-based parameterization.
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 Czerwiński et al. (2019).