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Characterization of AuxSNA(k, O(log n), poly) by modified cascading circuits

Determine how to modify the definition of semi-unbounded fan-in cascading circuits so as to obtain an exact circuit characterization for AuxSNAdepth,space,time(k, O(log n), n^{O(1)})_{FBS} for every k ≥ 1, extending the existing result that characterizes AuxSNAdepth,space,time(2k, O(log n), n^{O(1)})_{FBS}.

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Background

The main theorem establishes an equivalence between AuxSNA with depth 2k and k-cascading circuits. The construction hinges on the cascading-block length matching half the storage-tape rewrite depth bound.

The authors ask how to adapt the circuit model so that the same style of characterization holds for every depth k (including odd k), thereby closing the gap between 2k and k.

References

For future research, we wish to list eight key open questions associated with aux-$k$-sna's and $k$-cascading circuits. Our circuit characterization works for $AuxSNA\mathrm{depth,!space,!time}(2k,O(\log{n}),n{O(1)})_{FBS}$ with an arbitrary positive integer $k$. How can we modify the definition of cascading circuits to establish an exact characterization of $AuxSNA\mathrm{depth,!space,!time}(k,O(\log{n}),n{O(1)})_{FBS}$ for any $k\inN{+}$ in terms of circuit families?

— 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 3