The field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic

TL;DR

This paper extends the concept of automatic structures to ordinal-automatic structures, demonstrating that certain complex structures like the random graph and infinite integral domains are not ordinal-automatic below specific uncountable ordinals.

Contribution

It introduces the notion of ordinal-automatic structures and adapts Delhommé's relative-growth-technique to this setting, providing new non-automaticity results.

Findings

The random graph is not ordinal-automatic.

Infinite integral domains are not ordinal-automatic below ω₁+ω^ω.

Extension of automatic-structure theory to ordinal contexts.

Abstract

Recently, Schlicht and Stephan lifted the notion of automatic-structures to the notion of (finite-word) ordinal-automatic structures. These are structures whose domain and relations can be represented by automata reading finite words whose shape is some fixed ordinal $\alpha$. We lift Delhomm\'e's relative-growth-technique from the automatic and tree-automatic setting to the ordinal-automatic setting. This result implies that the random graph is not ordinal-automatic and infinite integral domains are not ordinal-automatic with respect to ordinals below $\omega_1+\omega^\omega$ where $\omega_1$ is the first uncountable ordinal.