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Turing degrees of multidimensional SFTs
"... In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2dimensional 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 comput ..."
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In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2dimensional 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 noncomputable 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 recursiveflavoured results about SFTs were proved in a Wang tile setting.
On Immortal Configurations in Turing Machines
, 2012
"... 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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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.