ONE DIMENSIONAL CELLULAR AUTOMATA(CA). By Bertrand Rurangwa bertrand LUT, 21May2010
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 () t ai k. 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 ( t) ( t1), ( t1),... ( t1),... ( t1) i ( ir) ( ir1) ( i) ( ir) a F a a a a F is called the automata rule.
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. jr ( t) ( t1) ai f ja( i j) jr
where the j are integer constants and thus f the function has a single integer as argument. Number of automata rules Consider a CA with K possible states per cell and a range r the different combinations are 2r 1. K
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. C t k 2 C 2 x
The discretization forward in time and symmetric in space reads tk. C C C 2C C ( x) ( t) ( t1) ( t1) ( t1) ( t1) i i 2 i1 i i1 j1 j1 C ( t1) j i j j1 ( t 1) f jci j j1
Fundamental differences: -The coefficients j in general are real numbers and not integers. -The number C j of states of is infinite.
C t 1 4 2 C 2 x Footer
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