  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.
