an:06246048
Zbl 1417.03241
Jeandel, Emmanuel; Vanier, Pascal
Turing degrees of multidimensional SFTs
EN
Theor. Comput. Sci. 505, 81-92 (2013).
00328390
2013
j
03D28 37B10 52C20
tilings; subshift of finite type; undecidability; \(\Pi_1^0\) classes; Turing degree
Summary: In this paper, we are interested in computability aspects of subshifts and in particular Turing degrees of two-dimensional subshifts of finite type (SFTs) (i.e., tilings). To be more precise, we prove that, given any \(\Pi_1^0\) class \(P\) of \(\{0,1\}^\mathbb N\), there is an SFT \(X\) such that \(P\times\mathbb Z^2\) is recursively homeomorphic to \(X\setminus U\), 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.