Tree-automatic scattered linear orders

Abstract

Abstract. We study tree-automatic linear orders on regular tree languages. We first show that there is no tree-automatic scattered linear order, and in particular no well-order, on the set of all finite labeled trees. This also follows from results of Gurevich-Shelah [8] and Carayol-Löding [4]. We then show that a regular tree language admits a tree-automatic scattered linear order if and only if all trees are included in a subtree of the full binary tree with finite tree-rank. As a consequence of this characterization, we obtain an algorithm which, given a regular tree language, decides if the tree language can be well-ordered by a tree automaton. Finally, we connect tree automata with automata on ordinals and determine sharp lower and upper bounds for tree-automatic well-orders on natural examples of regular tree languages. Our proofs use elementary techniques of automata theory.

The isomorphism problem for tree-automatic ordinals with addition

“…Delhommé proved that any tree-automatic ordinal is strictly less than ω ω ω [6]. Jain, Khoussainov, Schlicht and Stephan [12] connected tree-automatic ordinals with automata working on ordinal words [26] and provided an alternative proof of Delhommé's result. However, in contrast to word-automatic ordinals, it is unknown if the isomorphism problem for tree-automatic ordinals is decidable.…”

Isomorphisms of scattered automatic linear orders

“…[23]), our lower bound Π 0 1 for the rigidity problem leaves quite some room for improvements. Since the rank of a tree automatic linear order is properly below ω ω [17,16], the proof of [23] can be adapted to show that the isomorphism and the rigidity problems for tree automatic scattered linear orders both belong to Σ 0 ω ω . But we only have the lower bounds Π 0 1 and Π 0 2 , resp.…”

“…the surveys [28,1] as well as the list of open questions [19], for very recent results not covered by the mentioned articles, see e.g. [5,10,17,16].…”