Eliminating extra assumptions in the aux-k-sna characterization
Determine whether the circuit characterization of AuxSNAdepth,space,time(2k, O(log n), n^{O(1)}) with (frozen) blank sensitivity can be extended to auxiliary depth-k storage automata without imposing the weak depth-susceptibility and no frozen blank turn conditions; in particular, ascertain if these conditions can be relaxed or removed entirely while preserving the characterization.
References
For future research, we wish to list eight key open questions associated with aux-$k$-sna's and $k$-cascading circuits. To achieve our characterization, in essence, we have required two specific conditions of aux-$k$-sna's: weak depth-susceptibility and no frozen blank turn. As shown in Lemma \ref{relax-FBS}, these conditions are immediately induced from another notion of (frozen) blank sensitivity. We now wonder if these conditions can be further relaxed or eliminated completely.