Cellular Automata
From wikipedia:
“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,
geometry, numerical analysis, and many other
mathematical fields), physics (quantum mechanics,
hydrodynamics, and fluid dynamics), economics
(game theory), computer science (linear
programming, computer architecture, selfreplicating machines, stochastic computing), and
statistics. He is generally regarded as one of the
greatest mathematicians in modern history.[2]”
John von Neumann
1903 – 1957
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”
Self reproducing machines
3
Self reproducing machines
“Quines”
4
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
5
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).
A “glider”
Demo: http://golly.sourceforge.net
•
science –>
Netlogo models library: Computer
Cellular Automata –> Life
•
Go through code
•
See http://www.bitstorm.org/gameoflife/
•
See
http://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://rendellattic.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:
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:
Elementary cellular automata
One-dimensional, two states (black and white)
Rule:
Elementary cellular automata
One-dimensional, two states (black and white)
Rule:
http://www.youtube.com/watch?v=60P7717-XOQ
http://mathworld.wolfram.com/ElementaryCellularAutomaton.html
See Netlogo models library –> Computer Science –> Cellular Automata –>
CA 1D Elementary
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.)
ECA 110 is a universal computer
(Matthew Cook, 2002)
Rule:
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 randomlooking behavior
– Complex and random-looking behavior in nature
comes from simple programs.
• Natural systems can be modeled using cellular-automatalike 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.