Wadge Degrees of Infinitary Rational Relations

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).