AI 中文总结
研究三维向量加法系统可达性问题的复杂度，通过分层可泵性分析，证明该问题属于指数空间，此前已知算法是非初等的，此研究给出新结论。
AI 中文摘要
向量加法系统（VASS）可看作是操作固定数量（维度）非负计数器值的有限状态自动机。可达性问题，即询问是否存在从一个由状态和计数器值定义的配置到另一个配置的运行，一直是理论计算机科学中算法上的长期挑战。当维度作为输入一部分时，该问题在2021年被证明是阿克曼完备的。对于大于2的固定维度，特别是3维，可达性问题的确切复杂度仍不清楚。长期以来，三维VASS可达性问题的已知算法是非初等的，而最著名的下界仅是从二维继承的PSPACE难。最近的突破给出了该问题的首个初等上界2-指数空间。本文证明了三维VASS可达性问题属于指数空间。证明基于分层可泵性分析，得出两个配置间最短运行的双指数长度界。
英文摘要
A VASS can be viewed as a finite-state automaton manipulating a fixed number (called its dimension) of counters holding non-negative values. The reachability problem, asking whether there is a run from one configuration, defined by a state and values of the counters, to another configuration, has been a long-standing algorithmic challenge in theoretical computer science. When the dimension is part of the input, the problem has been shown to be ACKERMANN-complete in 2021. For fixed dimension greater than 2, and in particular for dimension 3, the exact complexity of the reachability problem remains unclear. For a long time the known algorithms for the 3-dimensional VASS reachability problem had been non-elementary, while the best known lower bound is merely PSPACE hardness inherited from dimension 2. A recent breakthrough in (Czerwiński, Jecker, Lasota, Orlikowski, ICALP 2025) gave the first elementary upper bound for the problem, namely 2-EXPSPACE. In this paper it is shown that the reachability problem in 3-VASS belongs to EXPSPACE. The proof is based on a hierarchical pumpability analysis, yielding a doubly-exponential length bound on the shortest runs between two configurations.
Comments47 pages
