On Recognizable Languages of Infinite Pictures

• Main Author: Finkel, Olivier
• Other Authors: Équipe de Logique Mathématique (ELM), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2004
• Subjects: Languages of infinite pictures; tiling systems; ordinal Büchi automaton; topological complexity; Borel and analytic sets; E-recognizable; A-recognizable; decision problems; [INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO]; [INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC]; [MATH.MATH-LO]Mathematics [math]/Logic [math.LO]; sami
• Online Access: https://hal.science/hal-00355793; https://hal.science/hal-00355793/document; https://hal.science/hal-00355793/file/rec-pictures-%2Berratum.pdf

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 audience In 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).