On Recognizable Tree Languages Beyond the Borel Hierarchy

TL;DR

This paper explores the topological complexity of non-Borel recognizable tree languages, demonstrating their completeness in the difference hierarchy and their relation to game tree languages, revealing deep complexity distinctions.

Contribution

It establishes the existence of highly complex tree languages within the difference hierarchy and links their complexity to game tree languages, advancing understanding of topological classifications.

Findings

Constructed $D_{oldsymbol{ ext{omega}^n}}(oldsymbol{ ext{ extSigma}}_1^1)$-complete tree languages.

Proved unambiguous B"uchi automaton languages are Borel.

Showed $W_{(i,k)}$ languages are not in any $D_eta(oldsymbol{ extSigma}_1^1)$ for $eta<oldsymbol{ extomega}^oldsymbol{ extomega}$.

Abstract

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer $n \geq 1$, there is a $D_{\omega^n}({\bf \Sigma}^1_1)$-complete tree language L_n accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous B\"uchi tree automaton must be Borel. Then we consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i, k)$. We prove that the $D_{\omega^n}({\bf \Sigma}^1_1)$-complete tree languages L_n are Wadge reducible to the game tree language $W_{(i, k)}$ for $k-i \geq 2$. In particular these languages $W_{(i, k)}$ are not in any class $D_{\alpha}({\bf \Sigma}^1_1)$ for $\alpha < \omega^\omega$.