On Recognizable Languages of Infinite Pictures

An erratum is added at the end of the paper: The supremum of the set of Borel ranks of Büchi recognizable languages of infinite pictures is not the first non recursive ordinal $\omega_1^{CK}$ but an ordinal $\gamma^1_2$ which is strictly greater than the ordinal $\omega_1^{CK}$. This follows from a result proved by Kechris, Marker and Sami (JSL 1989).International audienceIn a recent paper, Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the Büchi and Muller ones, firstly used for infinite words. The authors asked for comparing the tiling system acceptance with an acceptance of pictures row by row using an automaton model over ordinal words of length $\omega^2$. We give in this paper a solution to this problem, showing that all languages of infinite pictures which are accepted row by row by Büchi or Choueka automata reading words of length $\omega^2$ are Büchi recognized by a finite tiling system, but the converse is not true. We give also the answer to two other questions which were raised by Altenbernd, Thomas and Wöhrle, showing that it is undecidable whether a Büchi recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable)

Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class ${\bf \Sigma_{\alpha}^0}$ (respectively ${\bf \Pi_{\alpha}^0}$). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a ${\bf \Sigma_{1}^1}$-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation

Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class ${\bf \Sigma_{\alpha}^0}$ (respectively ${\bf \Pi_{\alpha}^0}$). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a ${\bf \Sigma_{1}^1}$-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation