Complex Cellular Automata Genaro On complexity of chaotic elementary cellular Abstract automaton with memory: Rule 126 Introduction Antecedents Chaotic ECA Rule 126 Mean field analysis Genaro J. Mart´ ınez Basins analysis ECA with memory http://uncomp.uwe.ac.uk/genaro/ ECA with memory on ϕ R 126 Solving some International Center of Unconventional Computing problems with Bristol Institute of Technology φ R 126 maj :4 Self-organization by University of the West of England, Bristol, United Kingdom. structure formation Coding gliders Summer Solstice 2009 International Conference on Coding gliders by basin representation Coding gliders since Discrete Models of Complex Systems de Bruijn diagrams Implementing basic Gda´ nsk University, Gda´ nsk, Poland functions Conclusions June 22–24, 2009 Final remarks Thanks
Complex Cellular Abstract Automata Genaro Abstract Introduction Antecedents Using Rule 126 elementary cellular automaton (ECA) we will Chaotic ECA Rule 126 demonstrate that a chaotic discrete system — when enriched Mean field analysis Basins analysis with memory – exhibits complex dynamics. To quantify ECA with memory complexity of Rule 126 ECA with memory we study what ECA with memory on ϕ R 126 types dynamics constructed in Rule 126’s evolution emerge Solving some problems with since mean field theory, basins and de Bruijn diagrams. Later φ R 126 maj :4 we will display its complex dynamics emerging selecting a Self-organization by structure formation kind of memory for analyse interactions between gliders and Coding gliders Coding gliders by stationary patterns implementing specific functions. basin representation Coding gliders since de Bruijn diagrams Implementing basic functions Conclusions Final remarks Thanks
Complex Cellular Objective and goal Automata Genaro In this talk we will display a simple tool to extract complex Abstract systems from a family of chaotic discrete dynamical system. Introduction Antecedents We will employ a technique — memory based rule analysis Chaotic ECA Rule of using past history of a system to construct its present 126 Mean field analysis state and to predict its future. Basins analysis ECA with memory ECA with memory on ϕ R 126 Solving some problems with φ R 126 maj :4 complex chaotic Self-organization by structure formation transformed to CA CA Coding gliders Coding gliders by basin representation Coding gliders since de Bruijn diagrams Implementing basic functions Conclusions Final remarks Thanks
Complex Cellular Cellular automata Automata Genaro Cellular automata (CA) are discrete dynamical systems Abstract Introduction evolving on an infinite regular lattice. Antecedents Chaotic ECA Rule Definition 126 A CA is a 4-tuple A = < Σ , u , ϕ, c 0 > evolving in Mean field analysis Basins analysis d -dimensional latice, where d ∈ Z + . Such that: ECA with memory ECA with memory on ◮ Σ represents the alphabet ϕ R 126 Solving some ◮ u the local connection, where, problems with φ R 126 maj :4 u = { x 0 , 1 ,..., n − 1: d | x ∈ Σ } , therefore, u is a neigborhood Self-organization by structure formation ◮ ϕ the local function, such that, ϕ : Σ u → Σ Coding gliders Coding gliders by basin representation ◮ c 0 the initial condition, such that, c 0 ∈ Σ Z Coding gliders since de Bruijn diagrams Implementing basic functions Also, the local function induces a global transition between Conclusions configurations: Final remarks Thanks Φ ϕ : Σ Z → Σ Z .
Complex Cellular Dynamics in one dimension Automata Genaro Abstract central cell boundary limit define a ring left neighbor right neighbor Introduction Antecedents Chaotic ECA Rule 126 Mean field analysis neigborhood Basins analysis evolution space Elemental CA (ECA) is defined as follow: ECA with memory ECA with memory on ϕ R 126 t Solving some • Σ = { 0 , 1 } problems with φ R 126 maj :4 • u = { x 1 , x 0 , x − 1 } such that x ∈ Σ t+ 1 Self-organization by structure formation • the local function ϕ : Σ 3 → Σ Coding gliders Coding gliders by • c 0 the initial condition is the first ring with t = 0 basin representation Coding gliders since de Bruijn diagrams Implementing basic t+n functions Conclusions Final remarks Thanks
Complex Cellular Wolfram’s classification Automata Genaro Wolfram defines his classification in simple rules Abstract [Wolfram86], known as ECA. Also, this classification is Introduction extended to n -dimension. Antecedents Chaotic ECA Rule Classes 126 Mean field analysis Basins analysis ◮ A CA is class I, if there is a stable state x i ∈ Σ, such ECA with memory that all finite configurations evolve to the homogeneous ECA with memory on ϕ R 126 configuration . Solving some problems with ◮ A CA is class II, if there is a stable state x i ∈ Σ, such φ R 126 maj :4 Self-organization by that any finite configuration become periodic. structure formation Coding gliders ◮ A CA is class III, if there is a stable state, such that for Coding gliders by basin representation some pair of finite configurations c i and c j with the Coding gliders since de Bruijn diagrams stable state, is decidable if c i evolve to c j , such that any Implementing basic functions configuration become chaotic. Conclusions Final remarks ◮ Class IV includes all CA also called complex CA . Thanks
Complex Cellular Wolfram’s classes Automata Genaro Abstract Introduction Antecedents Chaotic ECA Rule 126 Mean field analysis Basins analysis ECA with memory ECA with memory on ϕ R 126 Rule 32 Rule 15 Solving some problems with φ R 126 maj :4 Self-organization by structure formation Coding gliders Coding gliders by basin representation Coding gliders since de Bruijn diagrams Implementing basic functions Conclusions Final remarks Rule 90 Rule 110 Thanks Figure: Behavior classes in ECA: uniform, periodic, chaotic and complex respectively.
Complex Cellular The case of study: ECA Rule 126 Automata Genaro � 1 if 110 , 101 , 100 , 011 , 010 , 001 ϕ R 126 = 0 if 111 , 000 Abstract Introduction Antecedents Chaotic ECA Rule 126 Mean field analysis Basins analysis ECA with memory ECA with memory on ϕ R 126 Solving some problems with φ R 126 maj :4 Self-organization by structure formation Coding gliders Coding gliders by basin representation Coding gliders since de Bruijn diagrams Implementing basic functions Conclusions Figure: Chaotic ECA evolution rule 126. Initial density start with Final remarks Thanks a 66% on a ring of 356 cells to 187 generations.
Complex Cellular Mean field analysis Automata Mean field theory is a proven technique for discovering statistical Genaro properties of CA without analyzing evolution spaces of individual rules. Abstract In this way, it was proposed to explain Wolfram’s classes by probability Introduction theory, resulting in a classification based on mean field theory curve: Antecedents ◮ class I: monotonic, entirely on one side of diagonal; Chaotic ECA Rule 126 ◮ class II: horizontal tangency, never reaches diagonal; Mean field analysis Basins analysis ◮ class IV: horizontal plus diagonal tangency, no crossing; ECA with memory ◮ class III: no tangencies, curve crosses diagonal. ECA with memory on ϕ R 126 Thus for one dimension we have: Solving some problems with φ R 126 maj :4 k 2 r +1 − 1 Self-organization by structure formation X ϕ j ( X ) p v t (1 − p t ) n − v p t +1 = (1) Coding gliders j =0 Coding gliders by basin representation such that j is a number of relations from their neighborhoods and X the Coding gliders since de Bruijn diagrams combination of cells x i − r , . . . , x i , . . . , x i + r . n represents the number of Implementing basic functions cells in neighborhood, v indicates how often state one occurs in Moore’s Conclusions Final remarks neighborhood, n − v shows how often state zero occurs in the Thanks neighborhood, p t is a probability of cell being in state one, q t is a probability of cell being in state zero (therefore q = 1 − p ).
Complex Cellular Mean field polynomial for ϕ R 126 Automata Mean field curve confirms that probability of state ‘1’ in space-time Genaro configurations of ECA Rule 126 is 0.75 this probability of high densities of 1’s with its maximum point in 0.5. Abstract Rule 126 is chaotic because the curve cross the identity. The first unstable Introduction fixed point at the origin f = 0 show that given very small number of cells, all Antecedents they in state ’1’ will spread quickly on the lattice. The stable fixed point is Chaotic ECA Rule f = 0 . 6683, which represent ‘concentration’ of ‘1’s that diminish during 126 automaton development. Such stable fixed point hints on existence of Mean field analysis Basins analysis non-trivial periodic structures emerging on ECA Rule 126, as was confirmed ECA with memory using filters. ECA with memory on ϕ R 126 p t +1 = 3 p t q t Solving some problems with φ R 126 maj :4 Self-organization by ϕ R 126 p structure formation Coding gliders Coding gliders by basin representation Coding gliders since de Bruijn diagrams Implementing basic functions Conclusions Final remarks Thanks q
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