# An Upper Bound on the Complexity of Recognizable Tree Languages

Abstract : The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D_n({\bf\Sigma}^0_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^\omega$ into the class ${\bf\Delta}^1_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bf\Delta}^1_2$.
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Cited literature [32 references]

https://hal.archives-ouvertes.fr/hal-01128609
Contributor : Dominique Lecomte <>
Submitted on : Tuesday, March 10, 2015 - 9:48:50 AM
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### Identifiers

• HAL Id : hal-01128609, version 1
• ARXIV : 1503.02840

### Citation

Olivier Finkel, Dominique Lecomte, Pierre Simonnet. An Upper Bound on the Complexity of Recognizable Tree Languages. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), EDP Sciences, 2015, 49 (2), pp.121-137. ⟨hal-01128609⟩

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