Characterizations of periods of multidimensional shifts

TL;DR

This paper characterizes the sets of periods of multidimensional shifts of finite type, linking them to complexity classes, and explores related notions of periodicity and subshifts.

Contribution

It provides a precise complexity-theoretic characterization of periodic sets in multidimensional shifts of finite type and extends these characterizations to sofic and effective subshifts.

Findings

Sets of periods of SFTs are exactly the sets in class NE

Counting functions for periods are in class #E

Provides characterizations for sofic and effective subshifts

Abstract

We show that the sets of periods of multidimensional shifts of finite type (SFTs) are exactly the sets of integers of the complexity class $\NE$. We also show that the functions counting their number are the functions of #E. We also give characterizations of some other notions of periodicity. We finish the paper by giving some characterizations for sofic and effective subshifts.