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Altenbernd, Thomas and W\"ohrle have considered acceptance of languages ofinfinite two-dimensional words (infinite pictures) by finite tiling systems,with usual acceptance conditions, such as the B\"uchi and Muller ones [1]. Itwas proved in [9] that it is undecidable whether a B\"uchi-recognizablelanguage of infinite pictures is E-recognizable (respectively, A-recognizable).We show here that these two decision problems are actually $\Pi_2^1$-complete,hence located at the second level of the analytical hierarchy, and "highlyundecidable". We give the exact degree of numerous other undecidable problemsfor B\"uchi-recognizable languages of infinite pictures. In particular, thenon-emptiness and the infiniteness problems are $\Sigma^1_1$-complete, and theuniversality problem, the inclusion problem, the equivalence problem, thedeterminizability problem, the complementability problem, are all$\Pi^1_2$-complete. It is also $\Pi^1_2$-complete to determine whether a givenB\"uchi recognizable language of infinite pictures can be accepted row by rowusing an automaton model over ordinal words of length $\omega^2$.

Highly Undecidable Problems about Recognizability by Tiling Systems