Scope-Bounded Reachability in Valence Systems

TL;DR

This paper introduces a generalized scope-bounded reachability analysis for valence systems, extending previous work on multi-pushdown systems, and provides complexity results for various storage mechanisms.

Contribution

It generalizes scope-bounded reachability to valence systems, encompassing multiple storage models, and characterizes the computational complexity across different scenarios.

Findings

Reachability is decidable in PSPACE for all storage mechanisms in the framework.

The complexity landscape varies with scope bounds and input descriptions of storage mechanisms.

The approach unifies analysis for pushdowns, vector addition systems, and their combinations.

Abstract

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.