On Factor Universality in Symbolic Spaces

TL;DR

This paper explores the concept of universality in symbolic dynamics by studying factoring relations up to rescaling, demonstrating the existence of universal cellular automata in certain classes while establishing non-existence results in others.

Contribution

It introduces a new perspective on factoring relations up to rescaling and proves the existence of universal cellular automata within a broad class, contrasting with negative results for subshifts.

Findings

Existence of a cellular automaton that can simulate any other in a large class.

Negative results on the existence of universal objects in some classes of subshifts.

Analysis using recursion theory to establish non-universality in certain classes.

Abstract

The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T, the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes.