Plane-Walking Automata

TL;DR

This paper investigates classes of multidimensional subshifts defined by multihead finite automata, revealing a hierarchy that collapses at the third level and comparing these classes to SFTs and sofic shifts.

Contribution

It introduces a hierarchy of subshift classes based on the number of automaton heads and shows its collapse at the third level, also comparing with known shift classes.

Findings

Hierarchy collapses at the third level in all dimensions.

All co-recursively enumerable subshifts are obtained at the third level.

Open problem related to two-counter machines is proposed.

Abstract

In this article, we study classes of multidimensional subshifts defined by multihead finite automata, in particular the hierarchy of classes of subshifts defined as the number of heads grows. The hierarchy collapses on the third level, where all co-recursively enumerable subshifts are obtained in every dimension. We also compare these classes to SFTs and sofic shifts. We are unable to separate the second and third level of the hierarchy in one and two dimensions, and suggest a related open problem for two-counter machines.