ONE DIMENSIONAL CELLULAR AUTOMATA(CA). By Bertrand Rurangwa

Transcription

1 ONE DIMENSIONAL CELLULAR AUTOMATA(CA). By Bertrand Rurangwa bertrand LUT, 21May2010

2 Cellula automata(ca) OUTLINE - Introduction. -Short history. -Complex system. -Why to study CA. -One dimensional CA. bertrand LUT, 14Mary2010

3 Complex Systems - From the turbulence in fluids, to global weather patterns, to beautifully intricate galactic structures, to the complexity of living organisms.

4 Historical examples of ornamental art. bertrand LUT, 14Mary2010

5 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.

6 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.

7 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.

8 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.

9 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.

10 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.

11 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.

12 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.

13 -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.

14 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

15 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

16 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

17 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

18 Fundamental differences: -The coefficients j in general are real numbers and not integers. -The number C j of states of is infinite.

19 C t C 2 x Footer

20 MURAKOZE