Wadge Degrees of Infinitary Rational Relations

TL;DR

This paper establishes that 2-tape B"uchi automata have the same topological acceptance power as Turing machines with B"uchi conditions, revealing complex hierarchies and completeness results for infinitary rational relations.

Contribution

It demonstrates the equivalence of topological hierarchies for 2-tape B"uchi automata and Turing machines, and characterizes the Borel ranks of infinitary rational relations.

Findings

Equal topological power of 2-tape B"uchi automata and Turing machines.

Existence of $ ext{Sigma}^0_eta$-complete and $ ext{Pi}^0_eta$-complete relations for all recursive ordinals.

The supremum of Borel ranks exceeds the first non-recursive ordinal $oldsymbol{ ho^1_2}$.

Abstract

We show that, from the topological point of view, 2-tape B\"uchi automata have the same accepting power as Turing machines equipped with a B\"uchi acceptance condition. The Borel and the Wadge hierarchies of the class RAT_omega of infinitary rational relations accepted by 2-tape B\"uchi automata are equal to the Borel and the Wadge hierarchies of omega-languages accepted by real-time B\"uchi 1-counter automata or by B\"uchi Turing machines. In particular, for every non-null recursive ordinal $\alpha$, there exist some $\Sigma^0_\alpha$-complete and some $\Pi^0_\alpha$-complete infinitary rational relations. And the supremum of the set of Borel ranks of infinitary rational relations is an ordinal $\gamma^1_2$ which is strictly greater than the first non-recursive ordinal $\omega_1^{CK}$. This very surprising result gives answers to questions of Simonnet (1992) and of Lescow and Thomas (1988,1994).