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Coincidence of AuxSNA(k, O(log n), poly) with LOGkSNA_FBS and extension to LOGkSDA

Determine whether AuxSNAdepth,space,time(k, O(log n), n^{O(1)})_{FBS} equals LOGkSNA_{FBS}, the logspace many-one closure of (frozen) blank-sensitive k-sna languages, and investigate whether an analogous equality holds for LOGkSDA.

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Background

The main results characterize AuxSNA with even depth (under FBS and additional technical conditions) via cascading circuits and place these classes within P. The relationship between the auxiliary machine class and the corresponding LOG-closure for FBS machines remains unsettled.

An equality would mirror classical results for LOGCFL and auxiliary pushdown automata; extending it to deterministic storage automata would further strengthen the framework.

References

For future research, we wish to list eight key open questions associated with aux-$k$-sna's and $k$-cascading circuits. Does $AuxSNA\mathrm{depth,!space,!time}(k,O(\log{n}),n{O(1)})_{FBS}$ coincide with $\mathrm{LOG}k\mathrm{SNA}_{FBS}$? If so, can we expand it to the case of $\mathrm{LOG}k\mathrm{SDA}$?

— 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 7