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Separations between LOG(k) levels and containment in SC^k

Determine whether LOG(k+1) differs from LOG(k) and whether LOG(k) is properly contained in SC^k for all integers k ≥ 1, where LOG(k) denotes the closure under logspace many-one reductions of languages recognized by depth-k storage automata and SC^k is the k-th level of Steve’s class SC.

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Background

The paper studies deterministic and nondeterministic depth-k storage automata and associated closure classes LOG(k). For deterministic auxiliary machines, it was shown previously that LOG(k) sits between LOGDCFL and SCk. For nondeterministic variants, the paper proves LOGCFL ⊆ LOG(k) ⊆ NSCk.

Despite these inclusions, the authors emphasize that fundamental structural questions about the LOG(k) hierarchy and its relation to SCk remain unresolved. This open problem asks for definitive separations or containments.

References

For example, we do not know whether $LOG{(k+1)}$ is different from $LOG{k}$ or whether $LOG{k}$ is properly contained in $\mathrm{SC}k$.

— Nondeterministic Auxiliary Depth-Bounded Storage Automata and Semi-Unbounded Fan-in Cascading Circuits (2412.09186 - Yamakami, 12 Dec 2024) in Section 1.1 (The Language Families kSDA and LOGkSDA)