Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable

TL;DR

This paper proves that it is decidable whether a tree automatic scattered linear ordering is also word automatic, and provides a method to compute the word automatic presentation if it exists.

Contribution

It establishes the decidability of recognizing word automaticity within tree automatic scattered linear orderings and offers a constructive approach.

Findings

Decidability of the word automaticity problem for tree automatic scattered linear orderings.

Existence of a computable word automatic presentation from a tree automatic one when applicable.

Advancement in understanding the relationship between tree and word automatic structures.

Abstract

A tree automatic structure is a structure whose domain can be encoded by a regular tree language such that each relation is recognisable by a finite automaton processing tuples of trees synchronously. Words can be regarded as specific simple trees and a structure is word automatic if it is encodable using only these trees. The question naturally arises whether a given tree automatic structure is already word automatic. We prove that this problem is decidable for tree automatic scattered linear orderings. Moreover, we show that in case of a positive answer a word automatic presentation is computable from the tree automatic presentation.