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.