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. arXiv:1805.08829 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.
翻译:古典的多米诺问题询问是否存在一个砖块, 其中没有给出任何被禁止的输入模式。 在本文中, 我们考虑到多米诺问题的周期性版本: 以一个被禁止的模式组成的家庭为输入, 它允许一个定期的平铺吗? 输入可能对应一个固定类型、 一个肮脏的子变或一个有效的子变的子变。 arXiv: 1805. 08829 证明, 以几何理由, 将二维的二维( $\ Pi_ 0 ⁇ 1$- complete) 放在二维中, 这个问题是双维的双维的双维( 几何一) 。 我们发现, 高维的多为 : 在有限类型的情况下, $d\ geq 4$, 3$d geq 3$ for sofic and effective 子变换位。 降使用一个子变换位的子, 通用计算和两个额外的控制周期。 这种复杂性跳得惊人, 有两个原因 : 第一, 它区分 2 和 3 3维次变位次变, 而大多数的属性在二维和高。