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).

翻译：状态向量加法系统（VASS）的几何维度概念由Leroux与Schmitz（2019）提出，并经Fu、Yang与Zheng（2024）形式化定义，其用于量化系统中循环效应所张成向量空间的维度。本文从几何维度视角探究VASS可达性问题，揭示了其与传统维度参数化方法的本质差异。特别地，我们通过扩展Czerwiński等人（2019）最初提出的泵引理技术，证明了几何维度为1维与2维的VASS可达性问题均具有PSPACE完备性。