Summary: The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this paper, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift.
[A. Grandjean et al., LPIcs – Leibniz Int. Proc. Inform. 107, Article 128, 13 p. (2018; Zbl 1499.68109)] proved that this problem is co-recursively enumerable (\(\Pi_0^1\)-complete) in dimension 2 for geometrical reasons. We show that it is much harder, namely analytic (\(\Sigma_1^1\)-complete), in higher dimension: \(d\geq 4\) in the finite type case, d \(\geq 3\) for sofic and effective subshifts. The reduction uses a subshift embedding universal computation and two additional dimensions to control periodicity. This complexity jump is surprising for two reasons: first, it separates 2- and 3-dimensional subshifts, whereas most subshift properties are the same in dimension 2 and higher; second, it is unexpectedly large.
For the entire collection see [Zbl 1482.68007].