Communication Complexity and Intrinsic Universality in Cellular Automata

TL;DR

This paper introduces necessary conditions for cellular automata to be intrinsically universal, using communication complexity to analyze their computational power and demonstrate limitations of certain classes.

Contribution

It develops a framework connecting communication complexity with cellular automata universality, providing new tools to prove non-universality.

Findings

Several classes of cellular automata are shown not to be intrinsically universal.

Communication complexity effectively distinguishes between different computational capabilities of cellular automata.

The approach simplifies the study of universality by focusing on natural problems related to cellular automata dynamics.

Abstract

The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most of the cases. In this article, we introduce necessary conditions for a cellular automaton to be "universal", according to a precise notion of simulation, related both to the dynamics of cellular automata and to their computational power. This notion of simulation relies on simple operations of space-time rescaling and it is intrinsic to the model of cellular automata. Intrinsinc universality, the derived notion, is stronger than Turing universality, but more uniform, and easier to define and study. Our approach builds upon the notion of communication complexity, which was primarily designed to study parallel programs, and thus is, as we show in this article, particulary well suited to the study of cellular automata: it allowed to show, by studying natural problems on the dynamics of cellular automata, that several classes of cellular automata, as well as many natural (elementary) examples, could not be intrinsically universal.