The Isomorphism Relation Between Tree-Automatic Structures

TL;DR

This paper studies the isomorphism problem for omega-tree-automatic structures, showing it is independent of ZFC axioms and not classified within certain descriptive set-theoretic hierarchies.

Contribution

It proves the isomorphism problem for various omega-tree-automatic structures is independent of ZFC and not a Σ₂¹ or Π₂¹ set, revealing its complex logical nature.

Findings

Isomorphism relation for omega-tree-automatic boolean algebras is independent of ZFC.

The problem is not a Σ₂¹ or Π₂¹ set for these structures.

Results highlight the complexity of classifying omega-tree-automatic structures.

Abstract

An $\omega$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $\omega$-tree-automatic structures. We prove first that the isomorphism relation for $\omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $\omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a $\Sigma_2^1$-set nor a $\Pi_2^1$-set.