Tree-Automatic Well-Founded Trees

Details Tree-Automatic Well-Founded Trees Tree-Automatic Well-Founded Trees

2012-01-26

arxiv.org/abs/1201.5495v1

Computer Science

Logic in Computer Science

Scientific paper

We investigate tree-automatic well-founded trees. For this, we introduce a new ordinal measure for well-founded trees, called the embedding rank, briefly erank. The erank of a well-founded tree is always smaller than the ordinary (ordinal) rank of a tree. We also show that the ordinal rank of a well-founded tree of erank \alpha is strictly bounded by \omega (\alpha+1). For string-automatic well-founded trees, it was shown by Kuske, Liu, and Lohrey that the erank is always finite. Here, using Delhomme's decomposition technique for tree-automatic structures, we show that the erank of a tree-automatic well-founded tree is strictly below \omega^\omega. As a corollary, we obtain that the ordinal rank of a string-automatic (resp., tree-automatic) well-founded tree is strictly below \omega^2 (resp., \omega^\omega). The result for the string-automatic case nicely contrasts a result of Khoussainov and Minnes, saying that the rank of a string-automatic well-founded partial order reaches all ordinals below \omega^\omega. As second application of the erank, we show that the isomorphism problem for tree-automatic well-founded trees is complete (under Turing-reductions) for level \Delta^0_{\omega^\omega} of the hyperarithmetical hierarchy.

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