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 Π 0 1 class P of {0, 1} N there is a SFT X such that P ×Z 2 is recursively homeomorphic to X \U where U is a computable set of points. As a consequence, if P contains a computable member, P and X have the exact same set of Turing degrees. On the other hand, we prove that if X contains only non-computable members, some of its members always have different but comparable degrees. This gives a fairly complete study of Turing degrees of SFTs. Wang tiles have been introduced by Wang [Wang(1961)] to study fragments of first order logic. Independently, subshifts of finite type (SFTs) were introduced to study dynamical systems. From a computational and dynamical perspective, SFTs and Wang tiles are equivalent, and most recursive-flavoured results about SFTs were proved in a Wang tile setting.

Abstract We investigate the immortality problem for Turing machines and prove that there exists a Turing Machine that is immortal but halts on every recursive configuration. The result is obtained by combining a new proof of Hooper’s theorem [11] with recent results on effective symbolic dynamics.

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