A Hierarchy of Tree-Automatic Structures

We consider $\omega^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $\omega^n$ for some integer $n\geq 1$. We show that all these structures are $\omega$-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $\omega^2$-automatic (resp. $\omega^n$-automatic for $n>2$) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for $\omega^n$-automatic boolean algebras, $n > 1$, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a $\Sigma_2^1$-set nor a $\Pi_2^1$-set. We obtain that there exist infinitely many $\omega^n$-automatic, hence also $\omega$-tree-automatic, atomless boolean algebras $B_n$, $n\geq 1$, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].

💡 Research Summary

The paper introduces a new class of structures called ωⁿ‑automatic structures, where n ≥ 1, and investigates their relationship with ω‑tree‑automatic structures. An ωⁿ‑automatic structure is a relational structure whose domain and basic relations are recognized by automata that read words of ordinal length ωⁿ. These automata extend the classical Büchi and Muller automata by adding a transition mechanism for limit ordinals: at a limit step the next state is determined by the set of states that appear infinitely often before that step. The authors first establish basic closure properties of regular ωⁿ‑languages (emptiness, union, intersection, complement) and show that any ωⁿ‑automatic structure can be represented by a Muller or Rabin tree automaton, i.e., it is also ω‑tree‑automatic. Consequently, the class of ωⁿ‑automatic structures inherits the well‑known decidability of first‑order theories and closure under first‑order interpretations that are characteristic of automatic structures.

The central contribution concerns the isomorphism problem for several natural families of ωⁿ‑automatic structures: Boolean algebras, partial orders, rings (commutative and non‑commutative), and groups. For ω²‑automatic Boolean algebras (and more generally for ωⁿ‑automatic Boolean algebras with n > 2, as well as the other algebraic classes) the authors prove that the isomorphism relation is independent of ZFC. The proof proceeds by constructing, for each n ≥ 1, an infinite family of atomless Boolean algebras Bₙ that are pairwise isomorphic under the Continuum Hypothesis (CH) but pairwise non‑isomorphic under an alternative axiom AT (“almost trivial”). Since CH and AT are both consistent with ZFC and lead to opposite conclusions about the isomorphism types, ZFC alone cannot decide the isomorphism relation for these classes.

From this independence result the authors derive a complexity lower bound: the isomorphism problem for ωⁿ‑automatic Boolean algebras (n ≥ 2) and for the other algebraic classes mentioned is neither a Σ₂¹‑set nor a Π₂¹‑set. In other words, the problem lies strictly beyond the second level of the analytical hierarchy. This strengthens earlier work showing that the isomorphism problem for ω‑tree‑automatic structures is not Σ₁²‑definable; here the authors push the boundary to show that even the broader class of ωⁿ‑automatic structures exhibits the same high complexity.

The paper also clarifies the hierarchy between the various notions of automaticity. While ω‑tree‑automatic structures are known to be more expressive than ω‑automatic ones, the authors demonstrate that every ωⁿ‑automatic structure (for any finite n) can be simulated by a Muller or Rabin tree automaton, thus fitting neatly into the ω‑tree‑automatic framework. This embedding preserves all decidability and closure properties, confirming that the hierarchy collapses at the level of tree automata for the purposes of logical analysis.

Finally, the authors compare their results with prior work. Earlier papers (e.g.,