Cellular Automata

Cellular Automata Anthony Lora

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

What is an Automaton? • 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)

Cellular Automata • 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

A Simple CA • 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

Example • 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

Self-Reproducing Automata • 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

Turing Machine 101 • 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

From TM to CA • 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 • 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.

From TM to CA • 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.

Features of a 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 bedefined for a d-dimensionalgrid • Von Neumann Neighborhood • Moore Neighborhood • Input vs Output Neighbors • If both are of equal size, thenthe 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

Conway’s Game of Life • 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

Wireworld • 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 • 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

Elementary CA • Introduced by Stephen Wolfram in 1983 • Two states, 1-D, left and right neighbors • Below is the set of rules for Rule165 • Total of 256 elementary CA

Elementary CA • Often depicted a single black cell in the center for initial conditions • Rule 165:

Rule 110 • Capable of universal computation • Mirror: Rule 124, Compliment: 137, Both: 193

Firing Squad Problem • Proposed by John Myhill in 1957 • Solved by John McCarthy and Marvin Minskyand 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

McCarthy and Minsky’s Solution

Resources • “A Brief History of Cellular Automata” by PalashSarkar • “Cellular Automata Laboratory” by Rudy Rucker and John Walker • WolfromMathWorld

Cellular Automata