Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is verification of safety and liveness properties in this model; despite an extensive research effort, some basic problems remain open, which is exemplified by the complexity status of the reachability problem that is still not fully clarified. The liveness problems are known to be closely related to the reachability problem, and various structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem of determining whether a net is structurally live, i.e. whether there is an initial marking for which it is live, remained open for some time; e.g. Best and Esparza (Inf Process Lett 116(6):423–427, 2016. https://doi.org/10.1016/j.ipl.2016.01.011) emphasize this open question. Here we show that the structural liveness problem for Petri nets is ExpSpace-hard and decidable. In particular, given a net N and a semilinear set S, it is decidable whether there is an initial marking of N for which the reachability set is included in S; this is based on results by Leroux (28th annual ACM/IEEE symposium on logic in computer science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, IEEE Computer Society, pp 23–32, 2013. https://doi.org/10.1109/LICS.2013.7).

中文翻译：

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Petri 网的结构活性是 ExpSpace-hard 和可判定的

Place/transition Petri 网是一类分布式系统的标准模型，其可达性空间可能是无限的。深入研究的主题之一是验证该模型中的安全性和活性属性；尽管进行了广泛的研究工作，但一些基本问题仍然悬而未决，例如可达性问题的复杂状态仍未完全阐明。已知活性问题与可达性问题密切相关，并且已经研究了与活性相关的网络的各种结构特性。有点令人惊讶的是，确定网络是否在结构上是有效的问题的可判定性状态，即是否存在其有效的初始标记，在一段时间内保持开放；例如 Best 和 Esparza（Inf Process Lett 116(6):423–427, 2016. https://doi.org/10。1016/j.ipl.2016.01.011）强调这个开放性问题。在这里，我们表明 Petri 网的结构活性问题是 ExpSpace-hard 和可判定的。特别地，给定一个网络 N 和一个半线性集 S，可判定是否存在 N 的初始标记，其可达性集包含在 S 中；这是基于 Leroux 的结果（第 28 届 ACM/IEEE 计算机科学逻辑研讨会，LICS 2013，美国洛杉矶新奥尔良，2013 年 6 月 25-28 日，IEEE 计算机协会，2013 年第 23-32 页。https： //doi.org/10.1109/LICS.2013.7）。