The aperiodic Domino problem in higher dimension

TL;DR

This paper investigates the computational complexity of the aperiodic Domino problem in higher dimensions, revealing it to be significantly more complex than in two dimensions, with a jump to analytic completeness.

Contribution

It demonstrates that the aperiodic Domino problem is $oldsymbol{ ext{Σ}}_1^1$-complete in dimensions 3 and 4, showing a surprising increase in complexity compared to dimension 2.

Findings

The problem is $oldsymbol{ ext{Σ}}_1^1$-complete in higher dimensions.

Complexity jump separates 2D from higher-dimensional subshifts.

The reduction uses universal computation embedding and additional dimensions.

Abstract

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.