The emptiness problem for valence automata over graph monoids

Abstract

This work studies which storage mechanisms in automata permit decidability of the emptiness problem. The question is formalized using valence automata, an abstract model of automata in which the storage mechanism is given by a monoid. For each of a variety of storage mechanisms, one can choose a (typically infinite) monoid M such that valence automata over M are equivalent to (one-way) automata with this type of storage. In fact, many important storage mechanisms can be realized by monoids defined by finite graphs, called graph monoids. Examples include pushdown stacks, partially blind counters (which behave like Petri net places), blind counters (which may attain negative values), and combinations thereof.Hence, we study for which graph monoids the emptiness problem for valence automata is decidable. A particular model realized by graph monoids is that of Petri nets with a pushdown stack. For these, decidability is a long-standing open question and we do not answer it here.However, if one excludes subgraphs corresponding to this model, a characterization can be achieved. Moreover, we provide a description of those storage mechanisms for which decidability remains open. This leads to a model that naturally generalizes both pushdown Petri nets and the priority multicounter machines introduced by Reinhardt.The cases that are proven decidable constitute a natural and apparently new extension of Petri nets with decidable reachability. It is finally shown that this model can be combined with another such extension by Atig and Ganty: We present a further decidability result that subsumes both of these Petri net extensions.

Recent Advances on Reachability Problems for Valence Systems (Invited Talk)

“…In contrast, let SC + consist of all graphs that are transitive forests and contain a PVASS-graph. One can show that if Γ is not a transitive forest, then REACH(Γ ) is undecidable [38]. Thus, SC + is the class of graphs for which decidability remains open.…”

“…Until a few years ago, Open Problem 3.2 seemed out of reach. However, given the recent stunning resolution of the complexity of reachability in VASS [3,4,20] and the fact that decidability for SC ± is shown in [38] using a reduction to reachability in VASS with nested zero tests, there is hope to obtain new insights into Open Problem 3.2.…”

The complexity of bidirected reachability in valence systems

“…In this paper, we systematically characterize the decidability and complexity of bidirected reachability problem in the general algebraic framework of bidirected valence automata [26,40]. Valence automata consist of a finite-state machine and a (typically infinite) monoid that represents a storage mechanism.…”

“…Our classification follows that of the classification of (non-bidirected) reachability problems on valence automata [40]. There, Zetzsche characterizes a class of graphs that precisely capture PPNs with one Petri net place.…”

Coverability Is Undecidable in One-Dimensional Pushdown Vector Addition Systems with Resets

“…However, this greater expressivity comes with a price: the coverability problem for PVASS is only known to be decidable in dimension one [12]. This problem captures most of the decision problems of interest and in particular safety properties, and is the stumbling block in a classification for a large family of models combining pushdown stacks and counters [16].…”