Universal Cellular Automata and Class 4

TL;DR

This paper demonstrates that certain cellular automata rules, especially those in Wolfram's Class 4, are capable of universal computation by embedding a small universal Turing machine, highlighting the importance of initial conditions.

Contribution

It provides an embedding of Minsky's universal Turing machine into cellular automata, showing their computational universality and the dependence of class identification on initial conditions.

Findings

Class 4 CA rules can be universal Turing machines.

The classification of CA rules depends on initial conditions.

Universal CA rules are distributed among various Wolfram classes.

Abstract

Wolfram has provided a qualitative classification of cellular automata(CA) rules according to which, there exits a class of CA rules (called Class 4) which exhibit complex pattern formation and long-lived dynamical activity (long transients). These properties of Class 4 CA's has led to the conjecture that Class 4 rules are Universal Turing machines i.e. they are bases for computational universality. We describe an embedding of a ``small'' universal Turing machine due to Minsky, into a cellular automaton rule-table. This produces a collection of $(k=18,r=1)$ cellular automata, all of which are computationally universal. However, we observe that these rules are distributed amongst the various Wolfram classes. More precisely, we show that the identification of the Wolfram class depends crucially on the set of initial conditions used to simulate the given CA. This work, among others, indicates that a description of complex systems and information dynamics may need a new framework for non-equilibrium statistical mechanics.