Strictness of the alternation hierarchy for plane-walking automata

Prove that, for every integer n ≥ 1, the hierarchy of subshifts recognized by alternating plane-walking automata is strict, i.e., establish Σn-regular subshifts form a proper subset of Σn+1-regular subshifts, and similarly for all combinations of Π, Σ, and Δ classes.

Background

The paper defines a hierarchy of classes of subshifts recognized by plane-walking automata based on quantifier alternation: Σn, Πn, and Δn. The authors prove that Σ1 and Π1 are incomparable and strictly larger than the deterministic class Δ1, and they provide additional separations. They conjecture that this hierarchy continues to be strict at all levels.

The conjecture parallels known strict hierarchies for k-nested tree-walking automata and suggests broader structural properties of alternating walking automata on two-dimensional subshifts.