Cold dynamics in cellular automata: a tutorial

Abstract

This tutorial is about cellular automata that exhibit cold dynamics. By this we mean zero Kolmogorov-Sinai entropy, stabilization of all orbits, trivial asymptotic dynamics, or similar properties. These are essentially transient and irreversible dynamics, but they capture many examples from the literature, ranging from crystal growth to epidemic propagation and symmetric majority vote. A collection of properties is presented and discussed: nilpotency and asymptotic, generic or mu-variants, unique ergodicity, convergence, bounded-changeness, freezingness. They all correspond to the cold dynamics paradigm at various degrees, and we study their links and differences by key examples and results. Besides dynamical considerations, we also focus on computational aspects: we show how such cold cellular automata can still compute under their dynamical constraints, and what are their computational limitations. The purpose of this tutorial is to illustrate how the richness and complexity of the model of cellular automata are preserved under such strong constraints. By putting forward some open questions, it is also an invitation to look more closely at this cold dynamics territory, which is far from being completely understood.