Periodicity in tilings

TL;DR

This paper characterizes the sets of periods in tiling systems, showing they correspond to specific complexity classes, thereby linking tiling periodicity with computational complexity.

Contribution

It provides a precise characterization of periodicity sets in tiling systems, connecting them to complexity classes NSPACE(n) and coNSPACE(n).

Findings

Periodicity sets in tiling systems match languages in NSPACE(n) and coNSPACE(n)

Up to recoding, these sets are exactly characterized by these complexity classes

The work bridges tiling theory and computational complexity

Abstract

Tilings and tiling systems are an abstract concept that arise both as a computational model and as a dynamical system. In this paper, we characterize the sets of periods that a tiling system can produce. We prove that up to a slight recoding, they correspond exactly to languages in the complexity classes $\nspace{n}$ and $\cne$.