Cellular Automata

John von Neumann 1903 – 1957

“a Hungarian-American mathematician and

polymath who made major contributions to a vast

number of fields,[1] including mathematics (set

theory, functional analysis, ergodic theory, geo

metry, numerical analysis, and many other

mathematical fields), physics (quantum mechanics,

hydrodynamics, and fluid dynamics), economics

(game theory), computer science (linear progra

mming, computer architecture, self-replicating ma

chines, stochastic computing), and statistics. He

is generally regarded as one of the greatest

mathematicians in modern history.[2]”

From wikipedia:

Published report on the EDVAC “a computer architecture in which the data and the program are both

stored in the computer's memory in the same address space”

3

Self reproducing machines

4

Self reproducing machines

“Quines”

5

Self reproducing machines

Quotes

“Truth is much too complicated to allow anything but approximations.”

“You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics ....In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.”■Suggesting to Claude Shannon a name for his new uncertainty function, as quoted in Scientific AmericanVol. 225 No. 3, (1971), p. 180

“If people do not believe that mathematics is simple, it is only

because they do not realize how complicated life is.”

“ I am thinking about something much more important than

bombs. I am thinking about computers”.-John von Neumann,

1946

What is a cellular automaton?

• light bulbs pictures

• relation to Turing machines– “non-von-Neumann-style architecture”

• invented by von Neumann

• CAs and universal computation

What is a cellular automaton?

Circular (“toroidal”) boundary conditions

time = 1 time = 2

Conway talks about creating Life:http://www.youtube.com/watch?v=FdMzngWchDk

Example: Game of Life

(John Conway, 1970s)

• Neighborhood: 2 dimensional 3x3 neighborhood:

• Rules: – A dead cell with exactly three live neighbors

becomes a live cell (birth). – A live cell with two or three live neighbors

stays alive (survival). – In all other cases, a cell dies or remains dead

(overcrowding or loneliness).

Demo: http://golly.sourceforge.net

A “glider”

Netlogo models library: Computer science –> Cellular Automata –> Life

Go through code

See http://www.bitstorm.org/gameoflife/

Seehttp://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

• Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration?

• Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration?

• Relation to the Halting Problem.

• Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration?

• Relation to the Halting Problem.

• Answer: No.

• Is there a general way (a “definite procedure”) to predict the behavior of Life from a given initial configuration?

• Relation to the Halting Problem.

• Answer: No.

• Reason “Life is Universal.” http://rendell-attic.org/gol/tm.htm

Elementary cellular automata

One-dimensional, two states (black and white)

Elementary cellular automata

One-dimensional, two states (black and white)

Rule:

Rule:

Elementary cellular automata

One-dimensional, two states (black and white)

Rule:

Elementary cellular automata

One-dimensional, two states (black and white)

Elementary cellular automata

One-dimensional, two states (black and white)

Rule:

Elementary cellular automata

One-dimensional, two states (black and white)

Rule:

Elementary cellular automata

One-dimensional, two states (black and white)

Rule:

http://mathworld.wolfram.com/ElementaryCellularAutomaton.html

See Netlogo models library –> Computer Science –> Cellular Automata –> CA 1D Elementary

http://www.youtube.com/watch?v=60P7717-XOQ

Wolfram’s Four Classes of CA Behavior

• Class 1: Almost all initial configurations relax after a transient period to the same fixed configuration (e.g., all black).

• Class 2: Almost all initial configurations relax after a transient period to some fixed point or some temporally periodic cycle of configurations, but which one depends on the initial configuration

• Class 3: Almost all initial configurations relax after a transient period to chaotic behavior. (The term ``chaotic'‘ here refers to apparently unpredictable space-time behavior.)

• Class 4: Some initial configurations result in complex localized structures, sometimes long-lived.

Rule:

ECA 110 is a universal computer(Matthew Cook, 2002)

Wolfram’s numbering of ECA:

0 1 1 0 1 1 1 0 = 110 in binary

– Transfer of information: moving particles

From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

– Transfer of information: moving particles

From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

– Transfer of information: moving particles

– Integration of information from different spatial locations: particle collisions

From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

– Transfer of information: moving particles

– Integration of information from different spatial locations: particle collisions

From http://www.stephenwolfram.com/publications/articles/ca/86-caappendix/16/text.html

Outline of proof

1. Define “cyclic tag systems” and prove they are universal (they can emulate Turing machines).

2. Show ECA 110 can emulate a cyclic tag system.

Wolfram’s hypothesis:

All class 4 CAs can support universal computation

Outline of Wolfram’s A New Kind of Science

(from MM review, Science, 2002)• Simple programs can produce complex, and

random-looking behavior– Complex and random-looking behavior in nature

comes from simple programs.

• Natural systems can be modeled using cellular-automata-like architectures

• Cellular automata are a framework for understanding nature

• Principle of computational equivalence

Principle of Computational Equivalence

1. The ability to support universal computation is very common in nature.

2. Universal computation is an upper limit on the sophistication of computations in nature.

3. Computing processes in nature are almost always equivalent in sophistication.