2.11. Lecture 10: Cellular automatons

Before this class you should:

  • Read Think Complexity, Chapter 5

Before next class you should:

  • Have an excellent reading week!

  • Read Think Complexity, Chapter 6

Note taker: Haroon Shahbaz

2.12. Overview

This lecture introduces cellular automata which are a class of simple, discrete dynamical systems in which space is divided into a regular grid of cells and time advances in uniform steps. Each cell has a finite number of possible states. At every time step all cells update simultaneously according to a fixed rule that depends only on the current state of the cell and the states of its neighbours. Despite their simplicity, CA can generate a wide range of behaviour, from uniformity to chaos to structures capable of computation. We focus on one dimensional rules studied by Stephen Wolfram, look at some canonical examples, and discuss links to universality and fractals. Theoretical concepts

2.13. Introduction

What is it?

A cellular automaton is defined by

  • a lattice of cells (typically one or two dimensional);

  • a finite set of states for each cell;

  • a neighbourhood that specifies which neighbouring cells influence the update;

  • a local update rule that maps the neighbourhood pattern to a new state for the centre cell.

  • Model of simple, discrete systems: CA are models with local,

    deterministic update rules that can produce complex global behaviour from simple parts.

  • The simulation domain is divided into discrete cells; each cell

    has a finite set of possible states (e.g., {0,1} for binary CA).

  • Updates are synchronous: at each discrete time step every cell

    updates its state based on a fixed rule that depends on its neighbourhood.

2.14. One-dimensional cellular automata

In one dimension the cells form a linear array. A simple example is elementary CA in which each cell is binary (0 or 1) and the neighbourhood is the cell itself plus its immediate left and right neighbours (3-cell neighbourhood).

The next state of the central cell is determined by the pattern of the three cells in its neighbourhood; there are (2^{3}=8) possible patterns. Wolfram encoded these eight outputs in a single integer between 0 and 255, which is why we talk about rule 110, rule 18, etc.

2.15. Wolfram experiments

Wolfram ran many elementary CA and classified their behaviour into four broad classes:

  • Class 1: rapid convergence to a homogeneous state (all 0 or all 1).

  • Class 2: simple, stable or periodic structures.

  • Class 3: chaotic, seemingly random patterns.

  • Class 4: complex localized structures with rich interactions; conjectured to be capable of universal computation.

He summarised rules with a small lookup table that maps each possible neighbourhood pattern to the next state of the centre cell.

Selected example rules

  • Rule 50: produces repeating motifs; a useful illustration of how a simple update rule yields predictable structures.

  • Rule 38: exhibits irregular behaviour in places and has been discussed as a source of pseudo-randomness.

  • Rule 110: a canonical Class 4 rule, supports mobile localized structures (gliders or spaceships) and is capable of universal computation (it can simulate a Turing machine).

Spaceships are patterns that translate across the lattice while preserving their shape. A spaceship is characterised by its period (the number of time steps after which it repeats), displacement (number of cells moved), and velocity (displacement divided by period).

2.16. Turing machines

An abstract sequential machine with a finite-state head that reads and writes symbols on an (unbounded) tape and moves left or right according to a transition table. One way to connect Turing machines and CA is to view the tape and head as an augmented 1-D array; one cell encodes both the tape symbol and the head state. For each combination of machine state and current symbol, the transition table specifies the symbol to write, the next machine state, and the head movement (L or R).

A system is Turing-universal if it can simulate any Turing machine. Some cellular automata (including Rule 110) are universal, which means they can compute any computable function in principle. Universality is rare among the space of all rules; most CA belong to Classes 1-3.