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Journal Articles Fundamenta Informaticae Year : 2009

## Highly Undecidable Problems about Recognizability by Tiling Systems

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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 $\Pi_2^1$-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 $\Pi^1_2$-complete. It is also $\Pi^1_2$-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 $\omega^2$.

### Dates and versions

ensl-00340791 , version 1 (22-11-2008)

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• HAL Id : ensl-00340791 , version 1
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### Cite

Olivier Finkel. Highly Undecidable Problems about Recognizability by Tiling Systems. Fundamenta Informaticae, 2009, 91 (2), pp.305-323. ⟨ensl-00340791⟩

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