Decidability of emptiness for the SC+ class of valence automata

Determine whether the emptiness problem is decidable for valence automata whose storage mechanisms lie in the class SC+, i.e., those obtained by arbitrarily alternating the operations of building stacks and adding N-counters (with self-loops allowed).

Background

Within the hierarchy of valence automata storage mechanisms, classes SC− and SC± are known to have decidable emptiness. SC+ strictly generalizes these by allowing arbitrary alternation of stack-building and addition of nonnegative counters, making it more expressive.

The paper shows that several unboundedness-related properties are equivalent to emptiness whenever emptiness is decidable, but for SC+ the decidability of emptiness itself is currently unknown, leaving a central gap in the classification.