Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable
Martin Huschenbett

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.
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Taxonomy
Topicssemigroups and automata theory · Algorithms and Data Compression · Natural Language Processing Techniques
