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Scope-Bounded Reachability in Valence Systems
arXiv - CS - Formal Languages and Automata Theory Pub Date : 2021-08-02 , DOI: arxiv-2108.00963
Aneesh K. Shetty, S. Krishna, Georg Zetzsche

Multi-pushdown systems are a standard model for concurrent recursive programs, but they have an undecidable reachability problem. Therefore, there have been several proposals to underapproximate their sets of runs so that reachability in this underapproximation becomes decidable. One such underapproximation that covers a relatively high portion of runs is scope boundedness. In such a run, after each push to stack $i$, the corresponding pop operation must come within a bounded number of visits to stack $i$. In this work, we generalize this approach to a large class of infinite-state systems. For this, we consider the model of valence systems, which consist of a finite-state control and an infinite-state storage mechanism that is specified by a finite undirected graph. This framework captures pushdowns, vector addition systems, integer vector addition systems, and combinations thereof. For this framework, we propose a notion of scope boundedness that coincides with the classical notion when the storage mechanism happens to be a multi-pushdown. We show that with this notion, reachability can be decided in PSPACE for every storage mechanism in the framework. Moreover, we describe the full complexity landscape of this problem across all storage mechanisms, both in the case of (i) the scope bound being given as input and (ii) for fixed scope bounds. Finally, we provide an almost complete description of the complexity landscape if even a description of the storage mechanism is part of the input.

中文翻译:

价系统中的范围有界可达性

多下推系统是并发递归程序的标准模型,但它们存在不可判定的可达性问题。因此,已经有几个建议对它们的运行集进行欠逼近,以便在这种欠逼近中的可达性变得可判定。覆盖相对较高部分运行的一种此类欠近似是范围有界。在这样的运行中,每次推入堆栈 $i$ 后,相应的弹出操作必须在访问堆栈 $i$ 的有限次数内。在这项工作中,我们将这种方法推广到一大类无限状态系统。为此,我们考虑价系统模型,该模型由有限状态控制和由有限无向图指定的无限状态存储机制组成。该框架捕获下推、向量加法系统、整数向量加法系统及其组合。对于这个框架,我们提出了一个范围有界的概念,当存储机制恰好是一个多下推时,它与经典概念一致。我们表明,有了这个概念,可以在 PSPACE 中为框架中的每个存储机制决定可达性。此外,我们描述了这个问题在所有存储机制中的完整复杂性,在(i)范围界限作为输入和(ii)固定范围界限的情况下。最后,即使存储机制的描述是输入的一部分,我们也提供了对复杂性景观的几乎完整的描述。我们提出了一个范围有界的概念,当存储机制恰好是一个多下推时,它与经典概念一致。我们表明,有了这个概念,可以在 PSPACE 中为框架中的每个存储机制决定可达性。此外,我们描述了这个问题在所有存储机制中的完整复杂性,在(i)范围界限作为输入和(ii)固定范围界限的情况下。最后,即使存储机制的描述是输入的一部分,我们也提供了对复杂性景观的几乎完整的描述。我们提出了一个范围有界的概念,当存储机制恰好是一个多下推时,它与经典概念一致。我们表明,有了这个概念,可以在 PSPACE 中为框架中的每个存储机制决定可达性。此外,我们描述了这个问题在所有存储机制中的完整复杂性,在(i)范围界限作为输入和(ii)固定范围界限的情况下。最后,即使存储机制的描述是输入的一部分,我们也提供了对复杂性景观的几乎完整的描述。在 (i) 范围界限作为输入和 (ii) 固定范围界限的情况下。最后,即使存储机制的描述是输入的一部分,我们也提供了对复杂性景观的几乎完整的描述。在 (i) 范围界限作为输入和 (ii) 固定范围界限的情况下。最后,即使存储机制的描述是输入的一部分,我们也提供了对复杂性景观的几乎完整的描述。
更新日期:2021-08-03
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