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Scope-Bounded Reachability in Valence Systems

Authors Aneesh K. Shetty, S. Krishna , Georg Zetzsche

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Aneesh K. Shetty
  • Department of Computer Science & Engineering IIT Bombay, India
S. Krishna
  • Department of Computer Science & Engineering IIT Bombay, India
Georg Zetzsche
  • Max Planck Institute for Software Systems, Kaiserslautern, Germany

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Aneesh K. Shetty, S. Krishna, and Georg Zetzsche. Scope-Bounded Reachability in Valence Systems. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 29:1-29:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Formal languages and automata theory
  • multi-pushdown systems
  • underapproximations
  • valence systems
  • reachability


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