Structures Without Scattered-Automatic Presentation

TL;DR

This paper investigates the limitations of L-automatic structures for fixed linear orders L with finite condensation rank, establishing bounds on their complexity and separating them from tree-automatic structures.

Contribution

It proves bounds on the complexity of L-automatic structures based on the condensation rank of L and distinguishes L-automatic from tree-automatic structures.

Findings

No scattered linear order with high condensation rank is L-automatic.

Bounds on the height of L-automatic well-founded order trees.

The countable atomless boolean algebra is not L-automatic.

Abstract

Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L. Automata on finite words can be used to represent infinite structures, the so-called word-automatic structures. Analogously, for a linear order L there is the class of L-automatic structures. In this paper we prove the following limitations on the class of L-automatic structures for a fixed L of finite condensation rank 1+\alpha. Firstly, no scattered linear order with finite condensation rank above \omega^(\alpha+1) is L-\alpha-automatic. In particular, every L-automatic ordinal is below \omega^\omega^(\alpha+1). Secondly, we provide bounds on the (ordinal) height of well-founded order trees that are L-automatic. If \alpha is finite or L is an ordinal, the height of such a tree is bounded by \omega^{\alpha+1}. Finally, we separate the class of tree-automatic structures from that of L-automatic structures for any ordinal L: the countable atomless boolean algebra is known to be tree-automatic, but we show that it is not L-automatic.