Graph classification for decidable emptiness in valence automata

Determine the complete graph-theoretic characterization of all finite undirected graphs Γ (with possible self-loops) for which the emptiness problem for valence automata over the associated graph monoid MΓ is decidable. The goal is to identify precisely which storage mechanisms specified by Γ yield decidable emptiness for valence automata.

Background

Valence automata are finite-state automata equipped with a storage mechanism defined by a graph monoid MΓ. Their decidability landscape depends critically on the underlying graph Γ. Prior work shows Turing-completeness for certain induced subgraphs (e.g., P4 and C4) and decidability for specific subclasses, but a general classification of graphs with decidable emptiness remains unresolved.

This paper develops amalgamation systems and proves broad equivalences between emptiness and several unboundedness-related properties across valence automata. Nonetheless, the foundational question of which graphs Γ admit decidable emptiness persists as an open classification problem.