Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable

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.

💡 Research Summary

The paper addresses the fundamental question of whether a given tree‑automatic presentation of a scattered linear ordering can already be expressed as a word‑automatic presentation. Tree‑automatic structures encode their domain by a regular tree language and recognise relations with a synchronous finite automaton that reads tuples of trees in parallel. Word‑automatic structures are the special case where the trees are restricted to simple unary trees, i.e., words. While every word‑automatic structure is trivially tree‑automatic, the converse is not known in general, and the decidability of this “automaticity conversion” problem has been open for most classes of automatic structures.

The authors focus on scattered linear orderings, a well‑studied subclass of linear orders that contain no dense suborder. This restriction dramatically limits the combinatorial complexity of the order: any interval can be described by a finite set of “types” that capture the branching pattern of the underlying tree representation. Leveraging this property, the paper establishes three main contributions.

1. Normalization of Tree‑Automatic Presentations.
   They prove that any tree‑automatic presentation of a scattered linear order can be transformed, effectively and in elementary time, into a regular form: a tree language where all trees have a uniform depth and a bounded branching pattern. This transformation preserves the interpreted order and can be carried out by a constructive algorithm that rewrites the original automaton’s transition rules. The regular form serves as a bridge between the tree world and the word world because its structural regularity enables a finite classification of sub‑trees.

2. Decidable Word‑Automaticity Test.
   Using the regular form, the authors devise a decision procedure that determines whether the order admits a word‑automatic presentation. The key idea is to partition the domain into finitely many interval types based on the regular form’s branching patterns. For each type they define a corresponding word block—a finite word that encodes the entire subtree of that type. They then construct a synchronous word automaton whose states correspond to these types and whose transitions simulate the original tree automaton on the level of blocks. Because the number of types is finite (a direct consequence of scatteredness), the resulting automaton is finite and effectively computable.

   The final step checks whether the language of tuples accepted by this word automaton coincides with the original relation language. This is precisely a regular language equivalence problem, known to be decidable (PSPACE‑complete). Consequently, the whole procedure yields a decision algorithm for word‑automaticity of tree‑automatic scattered linear orders.

3. Effective Construction of a Word‑Automatic Presentation.
   When the test succeeds, the construction from step 2 can be turned around to produce an explicit word‑automatic presentation. The algorithm concatenates the word blocks associated with the interval types to obtain a regular word language that represents the domain. The transition table of the synchronous word automaton directly gives the recognisers for the order relation. Thus a word‑automatic presentation is not only guaranteed to exist but can be computed from the original tree‑automatic description.

The authors also analyse the computational complexity of the whole pipeline. The normalization step incurs at most exponential blow‑up, while the type enumeration and block construction are double‑exponential in the size of the input automaton. Overall, the decision problem lies in 2‑EXPTIME, which, although high, is the first non‑trivial upper bound for any non‑trivial class of automatic structures.

In the discussion, the paper acknowledges that the scatteredness assumption is essential: without it, the branching patterns can encode arbitrary dense orders, making the type‑finite property fail and the reduction to word blocks impossible. Nevertheless, the techniques introduced—regular‑form normalization, finite type decomposition, and block‑level synchronous automata—provide a promising template for tackling automaticity conversion in broader settings, such as partially ordered sets or non‑scattered linear orders.

In summary, the work settles the decidability of word‑automaticity for tree‑automatic scattered linear orderings, supplies an explicit construction method for the word‑automatic presentation, and opens a new line of inquiry into the interplay between tree‑ and word‑automatic representations in automatic structure theory.