  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 $\Pi_2^1$-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 $\Sigma^1_1$-complete, and the
universality problem, the inclusion problem, the equivalence problem, the
determinizability problem, the complementability problem, are all
$\Pi^1_2$-complete. It is also $\Pi^1_2$-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 $\omega^2$.
