Turing degrees of multidimensional subshifts

Abstract : In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2-dimensional SFTs (i.e. tilings). To be more precise, we prove that given any \pizu subset $P$ of $\{0,1\}^\NN$ there is a SFT $X$ such that $P\times\ZZ^2$ is recursively homeomorphic to $X\setminus U$ where $U$ is a computable set of points. As a consequence, if $P$ contains a recursive member, $P$ and $X$ have the exact same set of Turing degrees. On the other hand, we prove that if $X$ contains only non-recursive members, some of its members always have different but comparable degrees. This gives a fairly complete study of Turing degrees of SFTs.
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Emmanuel Jeandel, Pascal Vanier. Turing degrees of multidimensional subshifts. Theoretical Computer Science, Elsevier, 2013, http://dx.doi.org/10.1016/j.tcs.2012.08.027. ⟨10.1016/j.tcs.2012.08.027⟩. ⟨hal-00613165v3⟩

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