Turing degrees of multidimensional SFTs

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 Π1 class P of {0, 1} there is a SFT X such that P ×Z 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. Knowing whether a tileset can tile the plane with a given tile at the origin (also known as the origin constrained domino problem) was proved undecidable by Wang [Wang(1963)]. Knowing whether a tileset can tile the plane in the general case was proved undecidable by Berger [Berger(1964), Berger(1966)]. Understanding how complex, in the sense of recursion theory, the points of an SFT can be is a question that was first studied by Myers [Myers(1974)] in 1974. Building on the work of Hanf [Hanf(1974)], he gave a tileset with no computable tilings. Durand/Levin/Shen [Durand et al.(2008)Durand, Levin, and Shen] showed, 40 years later, how to build a tileset for which all tilings have high Kolmogorov complexity. A Π1 class (of sets) is an effectively closed subset of {0, 1} , or equivalently the set of oracles on which a given Turing machine does not halt. Π1 classes occur naturally in various areas in computer science and recursive mathematics, see e.g. [Cenzer and Remmel(1998), Simpson(2011a)] and the upcoming book [Cenzer and Remmel(2011)]. It is easy to see that any SFT is a Π1 class (up to a computable coding of ΣZ 2 into {0, 1}). This has various consequences. As an example, every non-empty SFT contains a point which is not Turing-hard (see ∗mail: [email protected] †mail: [email protected] 1 ha l-0 06 13 16 5, v er si on 3 1 Ju n 20 12 Durand/Levin/Shen [Durand et al.(2008)Durand, Levin, and Shen] for a selfcontained proof). The main question is how different SFTs are from Π1 classes. In the one-dimensional case, some answers to these questions were given by Cenzer/Dashti/King/Tosca/Wyman [Dashti(2008), Cenzer et al.(2008)Cenzer, Dashti, and King, Cenzer et al.(2012)Cenzer, Dashti, Toska, and Wyman]. The main result in this direction was obtained by Simpson [Simpson(2011b)], building on the work of Hanf and Myers: for every Π1 class S, there exists a SFT with the same Medvedev degree as S. The Medvedev degree roughly relates to the “easiest” Turing degree of S. What we are interested in is a stronger result: can we find for every Π1 class S a SFT which has the same Turing degrees? We prove in this article that this is true if S contains a computable point but not always when this is not the case. More exactly we build (Theorem 4.1) for every Π1 class S a SFT for which the set of Turing degrees is exactly the same as for S with the additional Turing degree of computable points. We also show that SFTs that do not contain any computable point always have points with different but comparable degrees (Corollary 5.11), a property that is not true for all Π1 classes. In particular there exist Π1 classes that do not have any points with comparable degrees. As a consequence, as every countable Π1 class contains a computable point, the question is solved for countable sets: the sets of Turing degrees of countable Π1 classes are the same as the sets of Turing degrees of countable sets of tilings. In particular, there exist countable sets of tilings with some non-computable points. This can be thought as a two-dimensional version of Corollary 4.7 in [Cenzer et al.(2012)Cenzer, Dashti, Toska, and Wyman]. This paper is organized as follows. After some preliminary definitions, we start with a quick proof of a generalization of Hanf, already implicit in Simpson [Simpson(2011b)]. We then build a very specific tileset, which forms a grid-like structure while having only countably many tilings, all of them computable. This tileset will then serve as the main ingredient to prove the result on the case of classes with a computable point in section 4. In section 5 we finally show the result on classes without computable points.