Highly Undecidable Problems about Recognizability by Tiling Systems
Olivier Finkel · Fundamenta Informaticae · 2009
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-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-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all П-complete. It is also П-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.