Ultimate Traces of Cellular Automata

TL;DR

This paper investigates the long-term behavior of cellular automata traces, providing conditions for certain trace sets and proving the undecidability of properties stable under ultimate coincidence.

Contribution

It introduces sufficient conditions for a set of infinite words to be a CA trace and establishes the undecidability of properties stable under ultimate coincidence.

Findings

Provided criteria for CA trace characterization

Proved undecidability of trace properties under ultimate coincidence

Analyzed the behavior of partial CA traces over subshifts

Abstract

A cellular automaton (CA) is a parallel synchronous computing model, which consists in a juxtaposition of finite automata (cells) whose state evolves according to that of their neighbors. Its trace is the set of infinite words representing the sequence of states taken by some particular cell. In this paper we study the ultimate trace of CA and partial CA (a CA restricted to a particular subshift). The ultimate trace is the trace observed after a long time run of the CA. We give sufficient conditions for a set of infinite words to be the trace of some CA and prove the undecidability of all properties over traces that are stable by ultimate coincidence.