Isomorphisms of scattered automatic linear orders

TL;DR

This paper establishes the undecidability of determining isomorphisms and automorphisms in scattered automatic linear orders, and precisely characterizes the complexity of automorphism existence.

Contribution

It proves undecidability results for isomorphisms and automorphisms in scattered automatic linear orders and identifies the exact arithmetical hierarchy level for automorphism existence.

Findings

Isomorphism of scattered tree automatic linear orders is undecidable.

Existence of automorphisms in scattered word automatic linear orders is undecidable.

Exact arithmetical hierarchy level for automorphism existence is determined.

Abstract

We prove that the isomorphism of scattered tree automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders are undecidable. For the existence of automatic automorphisms of word automatic linear orders, we determine the exact level of undecidability in the arithmetical hierarchy.