Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modeling and analysis of hardware, software, and database systems, as well as chemical, biological, and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and, currently, the best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from Symposium on Logic in Computer Science 2019. We establish a non-elementary lower bound, i.e., that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi, and other areas, which are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the current best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. We develop a construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R . At the heart of our proof is then a novel gadget, the so-called factorial amplifier that, assuming availability of counters that are zero testable and bounded by k , guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k . Repeatedly composing the factorial amplifier with itself by means of the former construction enables us to compute, in linear time, Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we, in fact, already establish hardness for h -exponential space for Petri nets with h + 13 counters.

中文翻译：

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Petri 网的可达性问题不是初级的

Petri 网，也称为向量加法系统，是一种长期建立的并发模型，在硬件、软件和数据库系统以及化学、生物和业务流程的建模和分析中具有广泛的应用。Petri 网的核心算法问题是可达性：从给定的初始配置中是否存在一系列有效的执行步骤来达到给定的最终配置。自 1960 年代以来，这个问题的复杂性一直悬而未决，它是验证理论中最突出的开放性问题之一。Mayr 在他的开创性 STOC 1981 工作中证明了可判定性，目前，发表的最佳上限是来自 2019 年计算机科学逻辑研讨会的 Leroux 和 Schmitz 的非原始递归 Ackermannian。我们建立了一个非基本的下界，即可达性问题需要一个时间和空间指数的塔。在这项工作之前，最好的下限一直是指数空间，这要归功于 1976 年的 Lipton。新的下限是一项重大突破，原因有几个。首先，它表明可达性问题比覆盖性（即状态可达性）问题要困难得多，覆盖性问题也无处不在，但自 1970 年代后期以来就已知它对于指数空间是完备的。其次，它意味着来自形式语言、逻辑、并发系统、过程演算和其他领域的大量问题，众所周知，这些问题都可以减少 Petri 网可达性问题，但也不是基本问题。第三，它使 Petri 网的两个关键扩展的可达性问题的当前最佳下界过时：分支和下推堆栈。我们开发了一种结构，它使用任意大的值对和比率R提供由以下限制的零可测试计数器R. 然后，我们证明的核心是一个新颖的小工具，即所谓的阶乘放大器，假设可用的计数器为零可测试且有界ķ, 保证产生任意大的值对，其比率恰好是的阶乘ķ. 通过前一种构造反复组合阶乘放大器，使我们能够在线性时间内计算模拟 Minsky 机器的 Petri 网，其计数器由指数塔包围，从而产生非基本下界。通过进一步完善这个方案，我们实际上已经建立了H-Petri 网的指数空间H+ 13 个计数器。