The reachability problem is a central decision problem for formal verification based on vector addition systems with states (VASS), which are equivalent to Petri nets and form one of the most studied and applied models of concurrency. Reachability for VASS is also inter-reducible with a plethora of problems from a number of areas of computer science. In spite of recent progress, the complexity of the reachability problem remains unsettled, and it is closely related to the lengths of shortest VASS runs that witness reachability. We consider VASS of fixed dimension, and obtain three main results. For the first two, we assume that the integers in the input are given in unary, and that the control graph of the given VASS is flat (i.e., without nested cycles). We obtain a family of VASS in dimension 3 whose shortest reachability witnessing runs are exponential, and we show that the reachability problem is NP-hard in dimension 7. These results resolve negatively questions that had been posed by the works of Blondin et al. in LICS 2015 and Englert et al. in LICS 2016, and contribute a first construction that distinguishes 3-dimensional flat VASS from 2-dimensional VASS. Our third result, by means of a novel family of products of integer fractions, shows that 4-dimensional VASS can have doubly exponentially long shortest reachability witnessing runs. The smallest dimension for which this was previously known is 14.

中文翻译：

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具有状态的固定维向量加法系统的可达性

可达性问题是基于带状态向量加法系统 (VASS) 的形式验证的核心决策问题，VASS 相当于 Petri 网，是研究和应用最多的并发模型之一。VASS 的可达性也可以与来自许多计算机科学领域的大量问题相互还原。尽管最近取得了进展，但可达性问题的复杂性仍未解决，它与见证可达性的最短 VASS 运行的长度密切相关。我们考虑固定维度的 VASS，并获得三个主要结果。对于前两个，我们假设输入中的整数以一元形式给出，并且给定 VASS 的控制图是平坦的（即，没有嵌套循环）。我们在维度 3 中获得了一个 VASS 族，其最短可达性见证运行是指数级的，并且我们表明可达性问题在维度 7 中是 NP 难的。这些结果解决了 Blondin 等人的作品提出的负面问题。在 LICS 2015 和 Englert 等人。在 LICS 2016 中，并贡献了第一个结构，将 3 维平面 VASS 与 2 维 VASS 区分开来。我们的第三个结果，通过一系列新的整数分数乘积，表明 4 维 VASS 可以具有双指数长的最短可达性见证运行。之前已知的最小维度是 14。在 LICS 2015 和 Englert 等人。在 LICS 2016 中，并贡献了第一个结构，将 3 维平面 VASS 与 2 维 VASS 区分开来。我们的第三个结果，通过一系列新的整数分数乘积，表明 4 维 VASS 可以具有双指数长的最短可达性见证运行。之前已知的最小维度是 14。在 LICS 2015 和 Englert 等人。在 LICS 2016 中，并贡献了第一个结构，将 3 维平面 VASS 与 2 维 VASS 区分开来。我们的第三个结果，通过一系列新的整数分数乘积，表明 4 维 VASS 可以具有双指数长的最短可达性见证运行。之前已知的最小维度是 14。