Highly Undecidable Problems about Recognizability by Tiling Systems

Olivier Finkel

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arXiv · 0811.3704

Highly Undecidable Problems about Recognizability by Tiling Systems

Olivier Finkel

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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 $Π_2^1$-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 $Σ^1_1$-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all $Π^1_2$-complete. It is also $Π^1_2$-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 $ω^2$.

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BibTeX RIS

Olivier Finkel. 2008-11-22. Highly Undecidable Problems about Recognizability by Tiling Systems. https://arxiv.org/abs/0811.3704

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