Cellular AutomataAnthony Lora

What are Automata? Von Neuman’s Machine A Simple CA CA ?= TM Features Examples

Outline

A self-operating machine or mechanism Automata Theory:

◦ Study of abstract, mathematical machines and systems.

◦ Includes:• Deterministic Finite Automata (DFA)• Nondeterministic Finite Automata (NFA) • Pushdown Automata (PDA)• Turing Machines (TM)• Cellular Automata (CA)

What is an Automaton?

Proposed by John von Neumann in the late 1940s to model self-reproducing organisms

If a machine were surrounded by “hardware soup,” could it create a working copy of itself?◦ Possible if the machine had a blueprint for itself◦ Make a copy of the blueprint and use it as

instructions for building Stanislaw Ulam suggested using cells to

represent the parts

Cellular Automata

One-dimensional array of cells Each cell has a state Time is divided into discrete units A function (the “local rule”) determines how

cell state changes◦ State change determined by the cell’s state and

the state of its two neighbors At each clock tick, the local rule is run on

the cells in parallel

A Simple CA

Possible states: 0, 1 Rules:

◦ If 1, become 0◦ If 0 and neighbors are 0, then remain 0◦ If 0 and at least one neighbor is 1, then become 1

Example

T = 2

0 1 0 1 0

T = 1

1 0 1 0 0

T = 0

0 1 0 0 0

T = 5

1 0 1 0 1

T = 4

0 1 0 1 0

T = 3

1 0 1 0 1

Von Neumann’s CA had cells capable of 29 different states, though it has been simplified since then.

Had to be capable of universal computation in order to be nontrivial◦ AKA – Turing-complete, or capable of simulating a

single-tape Turing machine

Self-Reproducing Automata

Four Parts◦ Tape - Two-way infinite, divided into cells, each

one holding a symbol from a finite set◦ Head - Can read symbols, move left and right one

cell, and update cell contents◦ State Register – Holds the state of the machine◦ Table – Holds the different instructions for the TM,

based on the symbol read and the state Update cell contents, if necessary Move left or right, if necessary Change state, if necessary

Turing Machine 101

The state of the cells has two components◦ Tape symbol from the equivalent TM◦ Whether the TM’s head is scanning the present

cell Rule 1:

◦ If the head is not scanning the cell, or either of its neighbors, the cell contents remain the same

From TM to CA

T = 1

… 0 / A 0 / B 0 / C …

T = 0

… 0 / A 0 / B 0 / C …

From TM to CA Rule 2:

◦ If the head is scanning the right cell and there is a left move, then the current cell will be scanned in the next step. Same goes for a left cell and right move.

◦ Example: At A, shift right.

T = 1

… 0 / A 1 / B 0 / C …

T = 0

… 1 / A 0 / B 0 / C …

Rule 3: ◦ If the present cell is being scanned, change

contents according to TM’s rules and stop scanning the cell, if necessary.

◦ Rule: At C, update to Z and shift left.

From TM to CA

Cell States◦ Set of all possible states must be finite◦ Usually exists a quiescent state◦ Polygeneous CA - Has cells whose states come

from different sets Geometry

◦ Can be a d-dimensional grid◦ Can be finite or infinite in any direction

Features of a CA

Neighborhood◦ Different neighborhoods can

be defined for a d-dimensional grid• Von Neumann Neighborhood• Moore Neighborhood

◦ Input vs Output Neighbors• If both are of equal size, then

the CA is balanced

Features of a CA

Local Rules◦ Hybrid CA – Each cell has its own local rules◦ Possible for cells to change their local rule at each

time step

Features of a CA

Made by John Horton Conway in 1970◦ Design a simple set of rules to model a population

Setup◦ States: 0 (non-living), 1 (living)◦ 2-dimensional with a Moore neighborhood◦ Rules:• Survival – If alive and has 2 or 3 living neighbors, then

remains alive• Birth – If non-living and has exactly 3 living neighbors,

then becomes alive • Death – If alive, becomes non-living if it has 0 or 1 living

neighbors or 4+ living neighbors• The Game of Life

Conway’s Game of Life

Introduced by Brian Silverman in 1987 as part of Phantom Fish Tank program◦ Uses four color states, a 2-D grid, and Moore

neighborhood◦ Rules

Color 0 is background and always stays that way Color 1 is an electron head and always becomes a

tail Color 2 is an electron tail and always turns into wire Color 3 is wire, which remains wire unless 1 or 2 of

the neighboring cells are electron heads, in which case it becomes an electron head as well

Wireworld

Wireworld

Can simulate digital logic circuits◦ 2002 – Nick Gardner demonstrates how Wireworld

can be used to multiply two 8-bit numbers◦ A computer that can count off prime numbers has

also been created

Wireworld

Introduced by Stephen Wolfram in 1983 Two states, 1-D, left and right neighbors Below is the set of rules for Rule 165

Total of 256 elementary CA

Elementary CA

Often depicted a single black cell in the center for initial conditions

Rule 165:

Elementary CA

Capable of universal computation Mirror: Rule 124, Compliment: 137, Both:

193

Rule 110

Proposed by John Myhill in 1957 Solved by John McCarthy and Marvin Minsky

and published in 1962 by Edward Moore Problem:

◦ A line of N soldiers stand in the idle state◦ Soldier on the far left (the general) gives the

signal to fire◦ Soldiers can only communicate with their left and

right neighbors◦ All soldiers must fire only once and at the same

time

Firing Squad Problem

Original solution takes 15 states and works in 3N time

Most optimal time is 2N-2 and takes thousands of states

A 6-state solution exists and a 4-state solution is impossible

Unknown whether a 5-state solution may exist

Firing Squad Problem

McCarthy and Minsky’s Solution

“A Brief History of Cellular Automata” by Palash Sarkar

“Cellular Automata Laboratory” by Rudy Rucker and John Walker

Wolfrom MathWorld

Resources