Unrecognizability of emptiness for 2-register automata

Determine whether the emptiness problem for 2-register automata remains unrecognizable when a 2-register automaton starts with the input length in its primary counter and increments of that counter never fail, in order to establish whether the main non-recursive trade-off theorem can be strengthened.

Background

The paper proves that the emptiness problem for two-counter deterministic automata remains unrecognizable even under the promises that the automata are bounded, always terminate, and obey a threshold. This result is then combined with a general theorem to derive non-recursive descriptional trade-offs between classes of machines with different computational power.

The authors leave unresolved whether the same unrecognizability result holds for 2-register automata, defined here as two-counter deterministic automata whose primary counter is initialized with the input value and whose primary-counter increments never fail. A positive answer would yield a slightly stronger version of the paper’s main theorem.