Cellular Automata and Applications

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Cellular Automata and Applications
Ajith Abraham Telephone Number (918)
5948188E-mail aa_at_cs.okstate.edu WWW
http//ajith.softcomputing.net
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Automata Theory An automaton is a system which
obtains, transforms, transmits and uses
information to perform its functions without
direct human participation. It is
self-operational (Lerner, 1972). The most
general form of deterministic automata is the
Turing machine. The notions of message, amount
of disturbance or 'noise', quantity of
information, coding technique, etc. in the study
of automata are related to cybernetics.
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Turing Machine
Machine Intelligence refers back to 1936, when
Alan M Turing proposed, the idea of universal
mathematics machine, a theoretical concept in the
mathematical theory of computability. Turing
and Emil Post independently proved that
determining the decidability of mathematical
propositions is equivalent to asking what sorts
of sequences of a finite number of symbols can be
recognized by an abstract machine with a finite
set of instructions. Such a mechanism is now
known as a Turing machine
Turing Machine http//www.turing.org.uk/turing/
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Turing Machine
A Turing machine is an abstract representation of
a computing device. It consists of a read/write
head that scans a (possibly infinite)
one-dimensional (bi-directional) tape divided
into squares, each of which is inscribed with a 0
or 1. Computation begins with the machine, in a
given "state", scanning a square. It erases what
it finds there, prints a 0 or 1, moves to an
adjacent square, and goes into a new state. This
behavior is completely determined by three
parameters (1) the state the machine is in, (2)
the number on the square it is scanning, and (3)
a table of instructions.
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Turing Machine
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Turing Machine
The table of instructions specifies, for each
state and binary input, what the machine should
write, which direction it should move in, and
which state it should go into. (E.g., "If in
State 1 scanning a 0 print 1, move left, and go
into State 3".) The table can list only
finitely many states, each of which becomes
implicitly defined by the role it plays in the
table of instructions. These states are often
referred to as the "functional states" of the
machine. A Turing machine, therefore, is more
like a computer program (software) than a
computer (hardware). Any given Turing machine can
be realized or implemented on an infinite number
of different physical computing devices.
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• Automata key concepts and principles
• Deterministic automata
• Statistical automata
• Memoryless automata
• Finite memory automata
• Infinite memory automata

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Deterministic automata The output is uniquely
determined by the input sequences that is, you
get a definite output given any input. The
behavior of such automata can be accurately
predicted if the transfer operator is known and
given in the form of a table of a logical
function, and if you also know the initial state
and the input sequence.
http//www.article19.com/shockwave/lifegame.htm
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Statistical Automata Random output sequences are
generated given any fixed input. The amount of
randomness can be set by applying a probability
of any output given the systems current state and
input sequence.
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Memoryless automataRecognizes only one input at
a time and produces output based on that input.
The output is not influenced by any additional
inputs which arrived before that one. The
reaction time of the automaton is constant for
all input signals. The internal state of such
an automaton is independent from any external
action.
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Finite Memory Automata It refers to the type of
automata where the group of output signals
generated at a given quantized time depends not
only on the signals applied at the same moment,
but also on those which arrived earlier. These
preceding external actions (or fragments of them)
are recorded in the automaton by a variation of
its internal state. The reaction of such an
automaton is uniquely determined by the group of
input signals which has arrived and by its
internal state at a given time. These factors
also determine the state into which the automaton
goes.
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Infinite memory automataIt refers to an
abstract circuit of a logical automaton which in
principle is suitable for realizing any
information processing algorithm. Turing
machine belongs to this type of automata.
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Cellular Automata (CA) Cellular Automata were
introduced in the late 1940s by John von Neumann
(von Neumann, 1966 Toffoli, 1987) and Stanislaw
Ulam. From the more practical point of view it
was introduced in the late 1960s when John
Horton Conway developed the Game of Life
(Gardner, 1970 Dewdney, 1989 Dewdney, 1990).
CAs are discrete dynamical systems and are
often described as a counterpart to partial
differential equations, which have the capability
to describe continuous dynamical systems.
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The meaning of discrete is, that space, time and
properties of the automaton can have only a
finite, countable number of states. The basic
idea is not to try to describe a complex system
from "above" - to describe it using difficult
equations Not to describe a complex system with
complex equations, but let the complexity emerge
by interaction of simple individuals following
simple rules.
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• Hence the essential properties of a CA are
• a regular n-dimensional lattice
• (n is in most cases of one or two dimensions),
  where each cell of this lattice has a discrete
  state
• a dynamical behavior, described by so called
  rules.
• These rules describe the state of a cell for the
  next time step, depending on the states of the
  cells in the neighborhood of the cell.

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Building Cellular Automata The Cell The basic
element of a CA is the cell. A cell is a kind
of a memory element and stores - to say it with
easy words - states. In the simplest case, each
cell can have the binary states 1 or 0. In more
complex simulation the cells can have more
different states.
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The Lattice These cells are arranged in a
spatial web - a lattice. The simplest one is
the one dimensional "lattice", meaning that all
cells are arranged in the form of a line
(string?). The most common CAs are built in
one or two dimensions. Whereas the one
dimensional CA has the big advantage, that it is
very easy to visualize. The states of one time
step are plotted in one dimension, and the
dynamic development can be shown in the second
dimension.
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A flat plot of a one dimensional CA hence shows
the states from time step 0 to time step n. A
two dimensional plot can evidently show only the
state of one time step. So visualizing the
dynamics of a 2D CA is by that reason more
difficult. By that reasons and because 1D CAs
are generally more easy to handle first of all
the one dimensional CAs have been exploited be
theoreticians.
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Neighborhoods To introduce dynamic into the
system, we have to add rules. The "job" of
these rules is to define the state of the cells
for the next time step. In cellular automata a
rule defines the state of a cell in dependence of
the neighborhood of the cell.
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von Neumann Neighborhood four cells, the cell
above and below, right and left from each cell
are called the von Neumann neighborhood of this
cell. The radius of this definition is 1, as only
the next layer is considered. Moore Neighborhood
the Moore neighborhood is an enlargement of the
von Neumann neighborhood containing the diagonal
cells too. In this case, the radius r 1 Extended
Moore Neighborhood equivalent to description of
Moore neighborhood above, but neighborhood
reaches over the distance of the next adjacent
cells. Hence the r2 (or larger). Margolus
Neighborhood a completely different approach
considers 2x2 cells of a lattice at once.
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Moore Neighborhood
von Neumann Neighborhood
Extended Moore Neighborhood
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• What to do with cells at borders? The influence
  depends on the size of the lattice.
• Example In a 10x10 lattice about 40 of the
  cells are border cells, in a 100x100 lattice only
  about 4 of the cells are of that kind.
• Anyway, this problem must be solved. Two
  solutions of this problem are common
• Opposite borders of the lattice are "sticked
  together". A one dimensional "line" becomes
  following that way a circle, a two dimensional
  lattice becomes a torus.
• the border cells are mirrored the consequence
  are symmetric border properties.
• The more usual method is the possibility 1.

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Applying Rules An example of "macroscopic"
dynamic resulting from local interaction is "the
wave" in a - say soccer-stadium. Each person
reacts only on the "state" of his neighbor(s).
If they stand up, he will stand up too, and
after a short while, he sits down again. Local
interaction leads to global dynamic.
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Every group of states of the neighborhood cells
is related to a state of the core cell. E.g.
consider a one-dimensional CA a rule could be
"011 -gt x0x", what means that the core cell
becomes a 0 in the next time step (generation) if
the left cell is 0, the right cell is 1 and the
core cell is 1. Every possible state has to be
described. Totalistic Rules the state of the
next state core cell is only dependent upon the
sum of the states of the neighborhood cells. E.g.
if the sum of the adjacent cells is 4 the state
of the core cell is 1, in all other cases the
state of the core cell is 0. Legal Rules a
special kind of totalistic rules are the legal
rules. As it is not of advantageous in most cases
to use rules that produce a pattern from total
zero-state lattices (all cells in the automaton
are 0), Wolfram defined the so called legal rules
. These rules are a subset of all possible rules,
a selection of rules that produce no ones from
zero-state lattices.
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• Important properties
• CAs develop in space and time
• CA is a discrete simulation method. Hence Space
  and Time are defined in discrete steps.
• CA is built up from cells, that are
• lined up in a string for one-dimensional
  automata
• arranged in a two or higher dimensional lattice
  for two- or higher dimensional automata
• the number of states of each cell is finite
• the states of each cell are discrete
• all cells are identical
• the future state of each cell depends only of
  the current state of the cell and the states of
  the cells in the neighborhood
• the development of each cell is defined by so
  called rules

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The Behavior of CAs Universality and the
Turing-Stuff A system that is capable to do
universal computation is able to perform any
finite algorithm. Only a CA calculating for an
infinite period of time can be universal.
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Class 1 cellular automata After a finite
number of time-steps, class one automata tend to
achieve an unique state from (nearly) all
possible starting conditions. Class 2
cellular automata This type of automata usually
creates patterns that repeat periodically
(typical with small periods) or are stable.
Class 3 cellular automata From nearly all
starting conditions, this type of CAs lead to
non-periodic - chaotic patterns. The statistical
properties of these patterns and the statistical
properties of the starting patterns are almost
identical (after a sufficient period of
time). Class 4 cellular automata After finite
steps of time, this type of CAs usually "dies" -
the state of all cells becomes zero. Nevertheless
a few stable (periodic) figures are possible. One
popular example of an automaton of this type is
the Game of Life.
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CA Applications The Game of Life (GOL) was one
of the first "applications" showing that cellular
automata are capable of producing dynamic
patterns and structures. The GOL is "plays" on
a two dimensional lattice with binary cell
states, Moore neighborhood and arbitrary border
conditions. To be vivid a 1 can be interpreted
that the cell is "living", a 0 that the cell is
"dead".
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• a cell that is dead at the time step t, becomes
  alive at time t1 if exactly three of the eight
  neighboring cells at time t were alive.
• a cell that is alive at time t dies at time t1
  if at time t less than two or more than three
  cells are alive.

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Gas Models Rules base on 2x2 parts of the
lattice
"o" stands for a particle (or a billiard ball)
and a "." stands for an empty cell
After applying the rule and proceeding to the
next time step, the 2x2 raster is moved for one
block diagonally. Hence the movement of a
particle needs two time-steps.
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Gas simulation of collision of two particles
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