The recent years have seen remarkable progress in establishing the complexity of the reachability problem for vector addition systems with states (VASS), equivalently known as Petri nets. Existing work primarily considers the case in which both the VASS as well as the initial and target configurations are part of the input. In this paper, we investigate the reachability problem in the setting where the VASS is fixed and only the initial configuration is variable. We show that fixed VASS fully express arithmetic on initial segments of the natural numbers. It follows that there is a very weak reduction from any fixed such number-theoretic predicate (e.g. primality or square-freeness) to reachability in fixed VASS where configurations are presented in unary. If configurations are given in binary, we show that there is a fixed VASS with five counters whose reachability problem is PSPACE-hard.

翻译：近年来，在确定带状态向量加法系统（VASS，等价于佩特里网）的可达性问题的复杂度方面取得了显著进展。现有研究主要考虑VASS以及初始和目标配置均作为输入一部分的情形。在本文中，我们研究了VASS固定且仅初始配置可变的设定下的可达性问题。我们证明，固定VASS能够完全表达自然数初始片段上的算术运算。由此可得，存在从任意固定此类数论谓词（例如素性判定或无平方性判定）到固定VASS中可达性问题的极弱归约，其中配置以一元表示给出。若配置以二进制表示，我们证明存在一个具有五个计数器的固定VASS，其可达性问题是PSPACE难的。