The Rank of Tree-Automatic Linear Orderings

Author Martin Huschenbett

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Martin Huschenbett

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Martin Huschenbett. The Rank of Tree-Automatic Linear Orderings. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 586-597, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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. The finite condensation rank (FC-rank) of a linear ordering measures how far it is away from being dense. We prove that the FC-rank of every tree-automatic linear ordering is below omega^omega. This generalises Delhommé's result that each tree-automatic ordinal is less than omega^omega^omega. Furthermore, we show an analogue for tree-automatic linear orderings where the branching complexity of the trees involved is bounded.
  • tree-automatic structures
  • linear orderings
  • finite condensation rank
  • computable model theory


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