An Example of Pi^0_3-complete Infinitary Rational Relation

We give in this paper an example of infinitary rational relation, accepted by a 2-tape B\"{u}chi automaton, which is Pi^0_3-complete in the Borel hierarchy. Moreover the example of infinitary rational relation given in this paper has a very simple structure and can be easily described by its sections

On the Accepting Power of 2-Tape Büchi Automata

International audienceWe show that, from a topological point of view, 2-tape Büchi automata have the same accepting power than Turing machines equipped with a Büchi 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 Büchi 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 Théorie Descriptive, Ph. D. Thesis, Université 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]

On the Continuity Set of an omega Rational Function

In this paper, we study the continuity of rational functions realized by B\"uchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function F has at least one point of continuity and that its continuity set C(F) cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational Pi^0_2-subset of X^omega for some alphabet X is the continuity set C(F) of an omega-rational synchronous function F defined on X^omega.Comment: Dedicated to Serge Grigorieff on the occasion of his 60th Birthda