  We show that, from a topological point of view, 2-tape B\"uchi automata have
the same accepting power than Turing machines equipped with a B\"uchi
acceptance condition. In particular, we show that for every non null recursive
ordinal alpha, there exist some Sigma^0_alpha-complete and some
Pi^0_alpha-complete infinitary rational relations accepted by 2-tape B\"uchi
automata. This very surprising result gives answers to questions of W. Thomas
[Automata and Quantifier Hierarchies, in: Formal Properties of Finite automata
and Applications, Ramatuelle, 1988, LNCS 386, Springer, 1989, p.104-119], of P.
Simonnet [Automates et Th\'eorie Descriptive, Ph. D. Thesis, Universit\'e Paris
7, March 1992], and of H. Lescow and W. Thomas [Logical Specifications of
Infinite Computations, In: "A Decade of Concurrency", LNCS 803, Springer, 1994,
p. 583-621].
