Automatic Presentations of Structures
LCC '94 Selected Papers from the International Workshop on Logical and Computational Complexity
Automatic linear orders and trees
ACM Transactions on Computational Logic (TOCL)
Recurrent Reachability Analysis in Regular Model Checking
LPAR '08 Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning
Automatic presentations for finitely generated groups
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
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We investigate tree-automatic well-founded trees. For this, we introduce a new ordinal measure for well-founded trees, called ∞-rank. The ∞-rankof a well-founded tree is always bounded from above by the ordinary (ordinal) rank of a tree. We also show that the ordinal rank of a well-founded tree of ∞-rankα is smaller than ω·(α+1). For string-automatic well-founded trees, it follows from [16] that the ∞-rankis always finite. Here, using Delhommé's decomposition technique for tree-automatic structures, we show that the ∞-rankof a tree-automatic well-founded tree is strictly below ωω. As a corollary, we obtain that the ordinal rank of a string-automatic (resp., tree-automatic) well-founded tree is strictly below ω2 (resp., ωω). The result for the string-automatic case nicely contrasts a result of Delhommé, saying that the ranks of string-automatic well-founded partial orders reach all ordinals below ωω. As a second application of the ∞-rankwe show that the isomorphism problem for tree-automatic well-founded trees is complete for level $\Delta^0_{\omega^\omega}$ of the hyperarithmetical hierarchy (under Turing-reductions). Full proofs can be found in the arXiv-version [11] of this paper.