A Hierarchy of Tree-Automatic Structures

TL;DR

This paper explores the hierarchy of automata recognizing structures over ordinal words, showing their equivalence to tree-automatic structures and analyzing the complexity of their isomorphism problems.

Contribution

It establishes that all $oldsymbol{ extomega^n}$-automatic structures are $oldsymbol{ extomega}$-tree-automatic and investigates the independence of their isomorphism relations from ZFC.

Findings

All $oldsymbol{ extomega^n}$-automatic structures are $oldsymbol{ extomega}$-tree-automatic.

The isomorphism relation for certain $oldsymbol{ extomega^n}$-automatic structures is independent of ZFC.

Existence of infinitely many pairwise isomorphic or non-isomorphic $oldsymbol{ extomega^n}$-automatic boolean algebras under different set-theoretic assumptions.

Abstract

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].