Highly Undecidable Problems about Recognizability by Tiling Systems

  • Authors:

  • Olivier Finkel

  • Affiliations:

  • (Correspd.) Equipe Modèles de Calcul et Complexité Laboratoire de l'Informatique du Parallélisme CNRS et Ecole Normale Supérieure de Lyon 46, Allée d'Italie 69364 Lyon Ced ...

  • Venue:
  • Fundamenta Informaticae - Machines, Computations and Universality, Part II
  • Year:
  • 2009

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Abstract

Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Büchi andMuller ones, in [1]. It was proved in [9] that it is undecidable whether a Büchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually П$^{1}_{2}$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". We give the exact degree of numerous other undecidable problems for Büchi-recognizable languages of infinite pictures. In particular, the nonemptiness and the infiniteness problems are Σ$^{1}_{1}$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all П$^{1}_{2}$-complete. It is also П$^{1}_{2}$-complete to determine whether a given Büchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ω$^{2}$.