A characterization of $p$-automatic sequences as columns of linear cellular automata

TL;DR

This paper characterizes $p$-automatic sequences over finite fields as columns of spacetime diagrams of linear cellular automata with periodic initial conditions, linking automata theory with cellular automata dynamics.

Contribution

It establishes an equivalence between $p$-automatic sequences and columns of linear cellular automata, providing a new perspective on their structure and generation.

Findings

$p$-automatic sequences correspond to columns of linear cellular automata.

Subshifts from length-$p$ substitutions can be realized as factors of cellular automata.

The characterization bridges automata theory and cellular automata dynamics.

Abstract

We show that a sequence over a finite field $\mathbb F_q$ of characteristic $p$ is $p$-automatic if and only if it occurs as a column of the spacetime diagram, with eventually periodic initial conditions, of a linear cellular automaton with memory over $\mathbb F_q$. As a consequence, the subshift generated by a length-$p$ substitution can be realized as a topological factor of a linear cellular automaton.