On Local Symmetries And Universality In Cellular Autmata

TL;DR

This paper demonstrates that in certain symmetric families of cellular automata, the proportion of universal automata approaches one, extending previous results and showing the undecidability of universality within these classes.

Contribution

It extends the density results of universal cellular automata to well-known families with local symmetries and multiple states, and proves the undecidability of universality in these families.

Findings

Asymptotic density of universal CA is 1 in several symmetric families.

Results apply to well-known CA families like outer-totalistic CA.

Universality problem remains undecidable in these families.

Abstract

Cellular automata (CA) are dynamical systems defined by a finite local rule but they are studied for their global dynamics. They can exhibit a wide range of complex behaviours and a celebrated result is the existence of (intrinsically) universal CA, that is CA able to fully simulate any other CA. In this paper, we show that the asymptotic density of universal cellular automata is 1 in several families of CA defined by local symmetries. We extend results previously established for captive cellular automata in two significant ways. First, our results apply to well-known families of CA (e.g. the family of outer-totalistic CA containing the Game of Life) and, second, we obtain such density results with both increasing number of states and increasing neighbourhood. Moreover, thanks to universality-preserving encodings, we show that the universality problem remains undecidable in some of those families.