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Separations among LOGkSDA, LOGkSNA, and LOG(k+1)SNA

Establish whether LOGkSDA ≠ LOGkSNA and whether LOGkSNA ≠ LOG(k+1)SNA for every k ∈ N^+, where LOGkSDA and LOGkSNA denote the logspace many-one closures of languages recognized by deterministic and nondeterministic depth-k storage automata, respectively.

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Background

The paper introduces LOGkSNA as a nondeterministic counterpart to LOGkSDA and shows basic inclusions relative to LOGCFL and NSCk. However, strict separations between these closures and across k are not known.

This question asks for definitive separations between deterministic and nondeterministic closures as well as a hierarchy for LOGkSNA.

References

For future research, we wish to list eight key open questions associated with aux-$k$-sna's and $k$-cascading circuits. We have introduced $\mathrm{LOG}k\mathrm{SNA}$ based on $k$-sna's. Is it true that $\mathrm{LOG}k\mathrm{SDA} \neq \mathrm{LOG}k\mathrm{SNA}$ and $\mathrm{LOG}k\mathrm{SNA} \neq \mathrm{LOG}(k+1)\mathrm{SNA}$ for any number $k\inN{+}$?

— 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 6