2D cellular automata: dynamics and undecidability

TL;DR

This paper explores the dynamics of 2D cellular automata by introducing new concepts like quasi-expansivity and quasi-sensitivity, establishing properties similar to 1D automata, and proving the undecidability of certain properties.

Contribution

It introduces quasi-expansivity and quasi-sensitivity for 2D CA, extending classical results and proving the undecidability of closingness in 2D automata.

Findings

Quasi-expansivity shares properties with 1D expansivity.

A classical dichotomy theorem holds for 2D CA.

Closingness is undecidable for 2D CA.

Abstract

In this paper we introduce the notion of quasi-expansivity for 2D CA and we show that it shares many properties with expansivity (that holds only for 1D CA). Similarly, we introduce the notions of quasi-sensitivity and prove that the classical dichotomy theorem holds in this new setting. Moreover, we show a tight relation between closingness and openness for 2D CA. Finally, the undecidability of closingness property for 2D CA is proved.