An upper bound on the complexity of recognizable tree languages

Finkel, Olivier; Lecomte, Dominique; Simonnet, Pierre

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 49 (2015) no. 2, pp. 121-137.

The third author noticed in his 1992 Ph.D. thesis [P. Simonnet, Automates et théorie descriptive (1992).] that every regular tree language of infinite trees is in a class $⅁\left(D_n\left({\Sigma }_2^0\right)\right)$ for some natural number $n\ge 1$, where $⅁$ 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 ${\Delta }_2^1$, 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 ${\Delta }_2^1$.

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