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Depth-susceptible vs (frozen) blank-sensitive deterministic depth-k storage automata for k ≥ 3

Determine whether the language families recognized by depth-susceptible deterministic depth-k storage automata and by (frozen) blank-sensitive deterministic depth-k storage automata coincide for all k ≥ 3; that is, decide if kSDA_susceptible = kSDA_FBS for k ≥ 3.

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Background

For k = 2, DCFL is characterized both by depth-susceptible 2-sda’s (prior work) and by (frozen) blank-sensitive 2-sda’s (this paper). Whether these two restrictions remain equivalent for higher depths is not known.

This question asks to compare the two models’ expressive power for all depths k ≥ 3.

References

For future research, we wish to list eight key open questions associated with aux-$k$-sna's and $k$-cascading circuits. In Lemma 2.1, $DCFL$ is characterized by depth-susceptible 2-sda's, whereas we characterize $DCFL$ using (frozen) blank-sensitive 2-sda's in Proposition \ref{new-result-FBS}. What is the relationship between these two notions over $k$-sda's for any $k\geq3$? More specifically, is it true that $k\mathrm{SDA}{suseptible} = k\mathrm{SDA}{FBS}$ for any $k\geq3$?

— Nondeterministic Auxiliary Depth-Bounded Storage Automata and Semi-Unbounded Fan-in Cascading Circuits (2412.09186 - Yamakami, 12 Dec 2024) in Section 7 (Brief Discussions and Future Research Directions), Item 8