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Transferring strictness results from k-nested tree-walking automata to plane-walking automata via tree embeddings

Demonstrate that strictness results for k-nested tree-walking automata can be transferred to plane-walking automata by constructing subshifts that embed trees on a zero background and proving that allowing plane-walking runs to leave the tree does not increase recognition power, thereby obtaining strict Σk+1 separations for plane-walking automata.

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Background

The authors propose an approach to establish strictness of the plane-walking hierarchy by encoding trees into two-dimensional subshifts and requiring each embedded tree to belong to a language Lk that separates levels in the k-nested hierarchy.

They highlight a key technical obstacle—plane-walking automata can leave the tree—which might add power. They explicitly leave the feasibility of this transfer as an open question.

References

We believe that these results can be brought to plane-walking automata by considering subshifts where trees are drawn on a background on zeroes (this can be ensured with finitely many forbidden patterns), and every tree must belong to Lk, where Lk is a tree language that is Σk+1 but not Σk (alternatively, k-nested and not k-1-nested). This is not straightforward as plane-walking automata have the ability to walk out of the tree, which should not provide additional recognition power, but pumping arguments are very tedious for alternating plane-walking automata. We leave this as an open question for future research.

— Subshifts defined by nondeterministic and alternating plane-walking automata (2409.08024 - Menibus et al., 12 Sep 2024) in Section “Strict hierarchy and tree-walking automata”