Tree-Automatic Well-Founded Trees

We investigate tree-automatic well-founded trees. Using Delhomme's decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.

[1]  Frank Stephan,et al.  Automatic linear orders and trees , 2005, TOCL.

[2]  Alexander Kartzow.  First-Order Model Checking on Generalisations of Pushdown Graphs , 2012, ArXiv.

[3]  Julia A. Knight,et al.  Computable structures and the hyperarithmetical hierarchy , 2000 .

[4]  Julia F. Knight,et al.  Classification from a Computable Viewpoint , 2006, Bulletin of Symbolic Logic.

[5]  Markus Lohrey,et al.  The isomorphism problem on classes of automatic structures with transitive relations , 2013 .

[6]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[7]  Mia Minnes,et al.  Model-theoretic complexity of automatic structures , 2009, Ann. Pure Appl. Log..

[8]  Mikolaj Bojanczyk,et al.  Tree-Walking Automata , 2008, LATA.

[9]  Philip W. Carruth,et al.  Arithmetic of ordinals with applications to the theory of ordered Abelian groups , 1942 .

[10]  Anil Nerode,et al.  Automatic Presentations of Structures , 1994, LCC.

[11]  Anuj Dawar,et al.  How the World Computes Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012 , 2012 .

[12]  Markus Lohrey,et al.  Tree-Automatic Well-Founded Trees , 2012, CiE.

[13]  Markus Lohrey,et al.  Automatic structures of bounded degree revisited , 2011, J. Symb. Log..

[14]  Graham P. Oliver,et al.  Automatic Presentations for Finitely Generated Groups , 2005, STACS.

[15]  Achim Blumensath,et al.  Finite Presentations of Infinite Structures: Automata and Interpretations , 2004, Theory of Computing Systems.

[16]  Anthony Widjaja Lin,et al.  Recurrent Reachability Analysis in Regular Model Checking , 2008, LPAR.

[17]  Sasha Rubin,et al.  Automata-based presentations of infinite structures , 2011, Finite and Algorithmic Model Theory.

[18]  Denis R. Hirschfeldt,et al.  Realizing Levels of the Hyperarithmetic Hierarchy as Degree Spectra of Relations on Computable Structures , 2002, Notre Dame J. Formal Log..

[19]  Achim Blumensath,et al.  Automatic structures , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[20]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[21]  C. Delhommé.  Automaticité des ordinaux et des graphes homogènes , 2004 .

[22]  Uri Abraham,et al.  Hausdorff ’s theorem for posets that satisfy the finite antichain property , 1999 .

[23]  Martin Huschenbett.  Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable , 2012, CiE.

[24]  André Nies,et al.  Automatic structures: richness and limitations , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..