Highly Undecidable Problems about Recognizability by Tiling Systems

Altenbernd, Thomas and Wohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Buchi andMuller ones, in [1]. It was proved in [9] that it is undecidable whether a Buchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually P$^{1}_{2}$-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 Buchi-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 P$^{1}_{2}$-complete. It is also P$^{1}_{2}$-complete to determine whether a given Buchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ω$^{2}$.

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