Highly Undecidable Problems about Recognizability by Tiling Systems

Altenbernd, Thomas and W"ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the B"uchi and Muller ones [1]. It was proved in [9] that it is undecidable whether a B"uchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually ${\mathrm{\Pi }}_2^1$-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 B"uchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are ${\mathrm{\Sigma }}_1^1$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all ${\mathrm{\Pi }}_2^1$-complete. It is also ${\mathrm{\Pi }}_2^1$-complete to determine whether a given B"uchi recognizable language of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ${\omega }^2$.