Minimality, transitivity and sensitivity of non-uniform cellular automata

TL;DR

This paper explores the relationship between minimality, transitivity, and sensitivity in non-uniform cellular automata, showing that minimality does not necessarily imply sensitivity, unlike in uniform CA.

Contribution

It constructs a two-dimensional NUCA that is minimal and transitive but not sensitive, and proves that recurrent rule assignment makes transitivity imply sensitivity.

Findings

Constructed a NUCA that is minimal and transitive but not sensitive.

Showed that recurrent local rule assignment makes transitivity imply sensitivity.

Demonstrated the difference between uniform and non-uniform CA regarding sensitivity.

Abstract

Every transitive cellular automaton (CA) is sensitive to initial conditions. We study this implication in the more general context of non-uniform cellular automata (NUCA) with finitely many different local update rules assigned to cells. We construct a two-dimensional NUCA that is minimal -- and hence transitive -- but that is not sensitive to initial conditions. The construction is based on an odometer NUCA on $\{0,1,2\}^\mathbb{N}$ which is nearly uniform in the sense that only the first cell uses a different local rule. Then we show that if the assignment of local rules in the cells is recurrent then transitivity implies sensitivity.