Highly Undecidable Problems about Recognizability by Tiling Systems

Abstract

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 and Muller ones [1]. It was proved in [9] that it is undecidable whether a Büchi-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 Büchi-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 ${\Pi }_2^1$-complete. It is also ${\Pi }_2^1$-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 ${\omega }^2$.

to appear in a Special Issue of the journal Fundamenta Informaticae on Machines, Computations and Universality