one dimensional cellular...
TRANSCRIPT
ONE DIMENSIONAL CELLULAR
AUTOMATA(CA).
bertrand LUT, 21May2010
By Bertrand Rurangwa
Cellula automata(CA)
OUTLINE
- Introduction.
-Short history.
-Complex system.
-Why to study CA.
-One dimensional CA.
bertrand LUT, 14Mary2010
Complex Systems
- From the turbulence in fluids, to global
weather patterns, to beautifully intricate
galactic structures, to the complexity of living
organisms.
Historical examples of ornamental art.
bertrand LUT, 14Mary2010
Five generic characteristics(CA) :
• Discrete lattice of cells: the system substrate
consists of a one, two or three-dimensional
lattice of cells.
• Homogeneity: all cells are equivalent.
• Discrete states: each cell takes on one of a
finite number of possible discrete states.
• Local interactions: each cell interacts only
with cells that are in its local neighborhood.
• Discrete dynamics: at each discrete unit time,
each cell updates its current state according to
a transition rule taking into account the states
of cells in its neighborhood.
Why Study CA?
Four partially overlapping motivations for
studying CA :
• As powerful computation engines.
• As discrete dynamical system simulators.
• As conceptual vehicles for studying pattern
formation and complexity.
• As original models of fundamental.
As powerful computation engines.
- С A allow very efficient parallel computational
implementations to be made of lattice models
in physics and thus for a detailed analysis of
many concurrent dynamical processes in
nature.
As discrete dynamical system
simulators
- CA allow systematic investigation of complex
phenomena by embodying any number of
desirable physical properties. CA can be used
as laboratories for studying the relationship
between microscopic rules and macroscopic
behavior- exact computability ensuring that the
memory of the initial state is retained exactly
for arbitrarily long periods of time.
As conceptual vehicles for studying
pattern formation and complexity
- CA can be treated as abstract discrete
dynamical systems embodying intrinsically
interesting, and potentially novel, behavioral
features.
As original models of fundamental
- CA allow studies of radically new discrete
dynamical approaches to microscopic physics,
exploring the possibility that nature locally and
digitally processes its own future states.
One-dimensional cellular automata
- One-dimensional cellular automata consist
of a number of uniform cells arranged like
beads on a string. If not stated otherwise
arrays with finite number of cells and
periodic boundary conditions will be
investigated, i.e. the beads form a
necklace.
-The state of cell i at time t is referred to as . The finite number of possible states are
labelled by non-negative integers from 0 to
k -1.
The state of each cell develops in time by iteration of the map
F is called the automata rule.
( ) ( 1) ( 1) ( 1) ( 1)
( ) ( 1) ( ) ( ), ,... ,...t t t t t
i i r i r i i ra F a a a a
( )t
i ka
The state of the ith cell at the new time
level t depends only on the state of the ith
cell and the r (range) neighbors to the left
and right at the previous time level t- 1.
( ) ( 1)
( )
j rt t
i j i j
j r
a f a
where the are integer constants and thus f the function has a single integer as argument.
Number of automata rulesConsider a CA with K possible states
per cell and a range r the different
combinations are .
j
2 1rK
Cellular automata as a discretization of
partial differential equations
Lattice-gas cellular automata - a special
type of cellular automata are relatively new
numerical schemes to solve physical
problems ruled by partial differential
equations.
2
2
C Ck
t x
The discretization forward in time and
symmetric in space reads
( ) ( 1) ( 1) ( 1) ( 1)
1 12
.2
( )
t t t t t
i i i i i
t kC C C C C
x
1( 1)
1
jt
j i j
j
C
1( 1)
1
jt
j i j
j
f C
Fundamental differences:
-The coefficients in general are real
numbers and not integers.
-The number of states of is infinite.
j
jC
Footer
2
2
1
4
C C
t x
MURAKOZE