The Mirage of Universality in Cellular Automata

TL;DR

This paper surveys cellular automata, emphasizing that there is no single universal model of computational universality, highlighting differences in embedding efficient versus unbounded computations, and revealing hidden universality in simple automata.

Contribution

It provides a comprehensive overview of cellular automata's computational capabilities, distinguishing between various forms of universality and embedding methods.

Findings

Some automata embed efficient bounded computation.

Others embed unbounded but inefficient computation.

Strong universality can be hidden in simple automata.

Abstract

This note is a survey of examples and results about cellular automata with the purpose of recalling that there is no 'universal' way of being computationally universal. In particular, we show how some cellular automata can embed efficient but bounded computation, while others can embed unbounded computations but not efficiently. We also study two variants of Boolean circuit embedding, transient versus repeatable simulations, and underline their differences. Finally we show how strong forms of universality can be hidden inside some seemingly simple cellular automata according to some classical dynamical parameters.