DOCSLIB.ORG

Think Complexity.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Think Complexity.Pdf Think Complexity Allen B. Downey Beijing • Cambridge • Farnham • Köln • Sebastopol • Tokyo Think Complexity by Allen B. Downey Copyright © 2012 Allen Downey. All rights reserved. Printed in the United States of America. Published by O’Reilly Media, Inc., 1005 Gravenstein Highway North, Sebastopol, CA 95472. O’Reilly books may be purchased for educational, business, or sales promotional use. Online editions are also available for most titles (http://my.safaribooksonline.com). For more information, contact our corporate/institutional sales department: (800) 998-9938 or [email protected]. Editors: Mike Loukides and Meghan Blanchette Indexer: Allen Downey Production Editor: Kristen Borg Cover Designer: Karen Montgomery Proofreader: Katie DePasquale Interior Designer: David Futato Illustrator: Robert Romano March 2012: First Edition. Revision History for the First Edition: 2012-02-21 First release See http://oreilly.com/catalog/errata.csp?isbn=9781449314637 for release details. Think Complexity is available under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (http://creativecommons.org/licenses/by-nc-sa/3.0/legalcode). The author maintains an online version at http://thinkcomplex.com/thinkcomplexity.pdf. Nutshell Handbook, the Nutshell Handbook logo, and the O’Reilly logo are registered trademarks of O’Reilly Media, Inc. Think Complexity, the image of a black eagle, and related trade dress are trademarks of O’Reilly Media, Inc. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and O’Reilly Media, Inc., was aware of a trademark claim, the designations have been printed in caps or initial caps. While every precaution has been taken in the preparation of this book, the publisher and author assume no responsibility for errors or omissions, or for damages resulting from the use of the information con- tained herein. ISBN: 978-1-449-31463-7 [LSI] 1329856783 Table of Contents Preface ..................................................................... vii 1. Complexity Science ...................................................... 1 What Is This Book About? 1 A New Kind of Science 2 Paradigm Shift? 3 The Axes of Scientific Models 4 A New Kind of Model 6 A New Kind of Engineering 7 A New Kind of Thinking 8 2. Graphs ............................................................... 11 What’s a Graph? 11 Representing Graphs 13 Random Graphs 17 Connected Graphs 17 Paul Erdős: Peripatetic Mathematician, Speed Freak 18 Iterators 18 Generators 20 3. Analysis of Algorithms .................................................. 21 Order of Growth 22 Analysis of Basic Python Operations 24 Analysis of Search Algorithms 26 Hashtables 27 Summing Lists 31 pyplot 33 List Comprehensions 34 4. Small World Graphs .................................................... 37 Analysis of Graph Algorithms 37 iii FIFO Implementation 38 Stanley Milgram 39 Watts and Strogatz 40 Dijkstra 41 What Kind of Explanation Is That? 43 5. Scale-Free Networks .................................................... 45 Zipf’s Law 45 Cumulative Distributions 46 Continuous Distributions 48 Pareto Distributions 49 Barabási and Albert 51 Zipf, Pareto, and Power Laws 52 Explanatory Models 54 6. Cellular Automata ..................................................... 57 Stephen Wolfram 58 Implementing CAs 59 CADrawer 61 Classifying CAs 62 Randomness 63 Determinism 64 Structures 65 Universality 67 Falsifiability 69 What Is This a Model Of? 70 7. Game of Life .......................................................... 73 Implementing Life 74 Life Patterns 76 Conway’s Conjecture 77 Realism 77 Instrumentalism 79 Turmites 80 8. Fractals .............................................................. 81 Fractal CAs 82 Percolation 84 9. Self-Organized Criticality ................................................ 87 Sand Piles 87 Spectral Density 88 Fast Fourier Transform 90 iv | Table of Contents Pink Noise 91 Reductionism and Holism 92 SOC, Causation, and Prediction 95 10. Agent-Based Models ................................................... 97 Thomas Schelling 97 Agent-Based Models 98 Traffic Jams 99 Boids 100 Prisoner’s Dilemma 102 Emergence 103 Free Will 104 11. Case Study: Sugarscape ................................................ 107 The Original Sugarscape 107 The Occupy Movement 108 A New Take on Sugarscape 108 Pygame 108 Taxation and the Leave Behind 109 The Gini Coefficient 109 Results with Taxation 110 Conclusion 112 12. Case Study: Ant Trails .................................................. 115 Introduction 115 Model Overview 115 API Design 117 Sparse Matrices 118 wx 118 Applications 120 13. Case Study: Directed Graphs and Knots ................................... 121 Directed Graphs 121 Implementation 122 Detecting Knots 122 Knots in Wikipedia 124 14. Case Study: The Volunteer’s Dilemma .................................... 125 The Prairie Dog’s Dilemma 125 Analysis 126 The Norms Game 127 Results 128 Improving the Chances 129 Table of Contents | v A. Call for Submissions ................................................... 131 B. Reading List ......................................................... 133 Index ..................................................................... 135 vi | Table of Contents Preface Why I Wrote This Book This book is inspired by boredom and fascination: boredom with the usual presentation of data structures and algorithms, and fascination with complex systems. The problem with data structures is that they are often taught without a motivating context; the problem with complexity science is that it is usually not taught at all. In 2005, I developed a new class at Olin College where students read about topics in complexity, implement experiments in Python, and learn about algorithms and data structures. I wrote the first draft of this book when I taught the class again in 2008. For the third offering, in 2011, I prepared the book for publication and invited the students to submit their work in the form of case studies for inclusion in the book. I recruited nine professors at Olin to serve as a program committee and choose the re- ports that were ready for publication. The case studies that met the standard are in- cluded in this book. For the next edition, we invite additional submissions from readers (see Appendix A). Suggestions for Teachers This book is intended as a scaffold for an intermediate-level college class in Python programming and algorithms. My class uses the following structure: Reading Complexity science is a collection of diverse topics. There are many interconnec- tions, but it takes time to see them. To help students see the big picture, I give them readings from popular presentations of work in the field. My reading list and sug- gestions on how to use it are in Appendix B. Exercises This book presents a series of exercises; many of them ask students to reimplement seminal experiments and extend them. One of the attractions of complexity is that the research frontier is accessible with moderate programming skills and under- graduate mathematics. vii Discussion The topics in this book raise questions in the philosophy of science, and lend themselves to further reading and classroom discussion. Case studies In my class, we spend almost half the semester on case studies. Students participate in an idea generation process, form teams, and work for 6–7 weeks on a series of experiments, which they then present in the form of a publishable 4–6 page report. An outline of the course and my notes are available at https://sites.google.com/site/comp modolin. Suggestions for Autodidacts In 2009–10, I was a Visiting Scientist at Google, working in their Cambridge office. One of the things that impressed me about the software engineers I worked with was their broad intellectual curiosity and drive to expand their knowledge and skills. I hope this book helps people like them explore a set of topics and ideas they might not encounter otherwise, practice programming skills in Python, and learn more about data structures and algorithms (or review material that might have been less engaging the first time around). Some features of this book intended for autodidacts are: Technical depth There are many books about complex systems, but most are written for a popular audience. They usually skip the technical details, which is frustrating for people who can handle it. This book presents the mathematics and other technical content you need to really understand this work. Further reading Throughout the book, I include pointers to further reading, including original papers (most of which are available electronically), related articles from Wikipe- dia,1 and other sources. Exercises and (some) solutions For many of the exercises, I provide code to get you started, and solutions if you get stuck or want to compare your code to mine. 1. Some professors have an allergic reaction to Wikipedia, on the grounds that students may depend too heavily on an unreliable source. Since many of my references are Wikipedia articles, I want to explain my thinking. First, the articles on complexity science and related topics tend to be very good; second, they are written at a level that is accessible after you have read this book (but sometimes not before); and finally, they are freely available to readers all over the world. If there is a danger in sending readers to these references, it is not that they are unreliable, but that the
Load more

Recommended publications
  • Think Complexity: Exploring Complexity Science in Python
    Think Complexity Version 2.6.2 Think Complexity Version 2.6.2 Allen B. Downey Green Tea Press Needham, Massachusetts Copyright © 2016 Allen B. Downey. Green Tea Press 9 Washburn Ave Needham MA 02492 Permission is granted to copy, distribute, transmit and adapt this work under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License: https://thinkcomplex.com/license. If you are interested in distributing a commercial version of this work, please contact the author. The LATEX source for this book is available from https://github.com/AllenDowney/ThinkComplexity2 iv Contents Preface xi 0.1 Who is this book for?...................... xii 0.2 Changes from the first edition................. xiii 0.3 Using the code.......................... xiii 1 Complexity Science1 1.1 The changing criteria of science................3 1.2 The axes of scientific models..................4 1.3 Different models for different purposes............6 1.4 Complexity engineering.....................7 1.5 Complexity thinking......................8 2 Graphs 11 2.1 What is a graph?........................ 11 2.2 NetworkX............................ 13 2.3 Random graphs......................... 16 2.4 Generating graphs........................ 17 2.5 Connected graphs........................ 18 2.6 Generating ER graphs..................... 20 2.7 Probability of connectivity................... 22 vi CONTENTS 2.8 Analysis of graph algorithms.................. 24 2.9 Exercises............................. 25 3 Small World Graphs 27 3.1 Stanley Milgram......................... 27 3.2 Watts and Strogatz....................... 28 3.3 Ring lattice........................... 30 3.4 WS graphs............................ 32 3.5 Clustering............................ 33 3.6 Shortest path lengths...................... 35 3.7 The WS experiment....................... 36 3.8 What kind of explanation is that?............... 38 3.9 Breadth-First Search.....................
    [Show full text]
  • The Fantastic Combinations of John Conway's New Solitaire Game "Life" - M
    Mathematical Games - The fantastic combinations of John Conway's new solitaire game "life" - M. Gardner - 1970 Suche Home Einstellungen Anmelden Hilfe MATHEMATICAL GAMES The fantastic combinations of John Conway's new solitaire game "life" by Martin Gardner Scientific American 223 (October 1970): 120-123. Most of the work of John Horton Conway, a mathematician at Gonville and Caius College of the University of Cambridge, has been in pure mathematics. For instance, in 1967 he discovered a new group-- some call it "Conway's constellation"--that includes all but two of the then known sporadic groups. (They are called "sporadic" because they fail to fit any classification scheme.) Is is a breakthrough that has had exciting repercussions in both group theory and number theory. It ties in closely with an earlier discovery by John Conway of an extremely dense packing of unit spheres in a space of 24 dimensions where each sphere touches 196,560 others. As Conway has remarked, "There is a lot of room up there." In addition to such serious work Conway also enjoys recreational mathematics. Although he is highly productive in this field, he seldom publishes his discoveries. One exception was his paper on "Mrs. Perkin's Quilt," a dissection problem discussed in "Mathematical Games" for September, 1966. My topic for July, 1967, was sprouts, a topological pencil-and-paper game invented by Conway and M. S. Paterson. Conway has been mentioned here several other times. This month we consider Conway's latest brainchild, a fantastic solitaire pastime he calls "life". Because of its analogies with the rise, fall and alternations of a society of living organisms, it belongs to a growing class of what are called "simulation games"--games that resemble real-life processes.
    [Show full text]
  • Symbiosis Promotes Fitness Improvements in the Game of Life
    Symbiosis Promotes Fitness Peter D. Turney* Ronin Institute Improvements in the [email protected] Game of Life Keywords Symbiosis, cooperation, open-ended evolution, Game of Life, Immigration Game, levels of selection Abstract We present a computational simulation of evolving entities that includes symbiosis with shifting levels of selection. Evolution by natural selection shifts from the level of the original entities to the level of the new symbiotic entity. In the simulation, the fitness of an entity is measured by a series of one-on-one competitions in the Immigration Game, a two-player variation of Conwayʼs Game of Life. Mutation, reproduction, and symbiosis are implemented as operations that are external to the Immigration Game. Because these operations are external to the game, we can freely manipulate the operations and observe the effects of the manipulations. The simulation is composed of four layers, each layer building on the previous layer. The first layer implements a simple form of asexual reproduction, the second layer introduces a more sophisticated form of asexual reproduction, the third layer adds sexual reproduction, and the fourth layer adds symbiosis. The experiments show that a small amount of symbiosis, added to the other layers, significantly increases the fitness of the population. We suggest that the model may provide new insights into symbiosis in biological and cultural evolution. 1 Introduction There are two main definitions of symbiosis in biology, (1) symbiosis as any association and (2) symbiosis as persistent mutualism [7]. The first definition allows any kind of persistent contact between different species of organisms to count as symbiosis, even if the contact is pathogenic or parasitic.
    [Show full text]
  • Easychair Preprint a Game of Life on a Pythagorean Tessellation
    EasyChair Preprint № 5345 A Game of Life on a Pythagorean Tessellation Soorya Annadurai EasyChair preprints are intended for rapid dissemination of research results and are integrated with the rest of EasyChair. April 18, 2021 A Game of Life on a Pythagorean Tessellation Soorya Annadurai Microsoft R&D Pvt. Ltd. Bangalore, India [email protected] Abstract—Conway’s Game of Life is a zero-player game played 3) Births. Each empty cell adjacent to exactly three neigh- on an infinite square grid, where each cell can be either ”dead” bors is a birth cell. It will become alive in the next or ”alive”. The interesting aspect of this game arises when we generation. observe how each cell interacts with its neighbors over time. There has been much public interest in this game, and several In a cellular automaton of this type, a single cell may do variants have become popular. Research has shown that similar one of four things within a single time step[6]: If it was dead Games of Life can exist on hexagonal, triangular, and other tiled but becomes alive, we say that it is born. If it was alive and grids. Games of Life have also been devised in 3 dimensions. remains alive, we say that it survives. If it was alive and In this work, another Game of Life is proposed that utilizes a Pythagorean tessellation, with a unique set of rules. Several becomes dead, we say that it dies. And if it was dead and interesting life forms in this universe are also illustrated.
    [Show full text]
  • Emergence from Symmetry: a New Type of Cellular Automata
    Emergence from Symmetry: A New Type of Cellular Automata Zan Pan ∗ Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, P. R. China Abstract In the realm of cellular automata (CA), Conway’s Game of Life (Life) has gained the most fondness, due to its striking simplicity of rules but an impressive diversity of behavior. Based on it, a large family of models were investigated, e.g. Seeds, Replicator, Larger than Life, etc. They all inherit key ideas from Life, determining a cell’s state in the next generation by counting the number of its current living neighbors. In this paper, a different perspective of constructing the CA models is proposed. Its kernel, the Local Symmetric Distribution Principle, relates to some fundamental concepts in physics, which maybe raise a wide interest. With a rich palette of configurations, this model also hints its capability of universal computation. Moreover, it possesses a general tendency to evolve towards certain symmetrical directions. Some illustrative examples are given and physical interpretations are discussed in depth. To measure the evolution’s fluctuating between order and chaos, three parameters are introduced, i.e. order parameter, complexity index and entropy. In addition, we focus on some particular simulations and giving a brief list of open problems as well. 1 Introduction The conception of cellular automaton (CA) stems from John von Neumann’s research on self-replicating artificial systems with computational universality [22]. However, it was John Horton Conway’s fas- cinating Game of Life (Life) [15, 8] which simplified von Neumann’s ideas that greatly enlarged its influence among scientists.
    [Show full text]
  • The Game of Life Trad
    The Game of Life -“If I sometimes smoked one way [...] sometimes another, sometimes not at all, varied the way I dressed, was nervous, serene, ambitious, lazy, lecherous, gluttonous, ascetic – where would my `self ́ be? What would I achieve? It’s the way a man chooses to limit himself that determines his character. A man without habits, consistency, redundancy – and hence boredom – is not human. He’s insane.” -“And accepting these self- defeating limitations is mental health?” I said. Luke Rhinehart, The Dice Man, Harper-Collins, 1999, pp. 65-66. Gaël Charbau: A “retrospective”... there’s a word I never thought I’d use about your work at such an early stage. Maybe “introspective” is a better description of this short exploration of your work. The larousse dictionary defines “introspection” as: “methodical observation of personal states of consciousness and inner life.” What are your thoughts? Gilles Barbier: Some of the work is twenty-five years old. That’s half my life. The direction of my work, however, has been based on drift and retrospective narratives have difficulty mapping out this fluid form which has no backbone. It isn’t easy to recount twenty-five years of drifting on an artificial lake without currents, winds, wrecks and islands, without making what is a maiden voyage appear disordered and agitated. This is my relationship to art. It is easier to talk about the holes, the bananas, the leaks, the insatiable hunger, the engines cutting out, the desire for treasure, and the great absentee, time, which is manifest only through its effects. GC: I wanted to ask you about the holes theme.
    [Show full text]
  • Ramifying Feedback Networks, Cross-Scale Interactions, And
    Ramifying Feedback Networks, Nicholas M. Gotts* Cross-Scale Interactions, and Macaulay Land Use Research Institute Emergent Quasi Individuals in Conway’s Game of Life Keywords Feedback network, cross-scale interaction, cellular automata, emergence Abstract Small patterns of state 1 cells on an infinite, otherwise empty array of Conway’s game of Life can produce sets of growing structures resembling in significant ways a population of spatially situated individuals in a nonuniform, highly structured environment. Ramifying feedback networks and cross-scale interactions play a central role in the emergence and subsequent dynamics of the quasi population. The implications are discussed: It is proposed that analogous networks and interactions may have been precursors to natural selection in the real world. 1 Introduction There are small patterns of state 1 cells on an infinite, otherwise empty array of Conway’s game of Life [5], which manifest a long-sustained increase in structural and dynamic complexity, and produce sets of growing structures resembling in significant ways a group of spatially situated individual organisms interacting with a nonuniform environment. These quasi individuals’ interaction with the environment and each other, but not their internal makeup, appears lifelike: They grow (although having no metab- olism), affect their neighbors’ survival, and influence the appearance of new group members (although no genetic mechanism is involved). Ramifying (self-extending) feedback networks and cross-scale in- teractions are central to the emergence and subsequent dynamics of these quasi individuals. Conway’s game of Life (henceforth GoL) is a synchronous, deterministic, binary cellular automaton (CA), run on a two-dimensional array of rectangular cells (infinite, or finite and toroidal).
    [Show full text]
  • Symbiosis Promotes Fitness Improvements in the Game of Life
    Symbiosis Promotes Fitness Improvements in the Game of Life Peter D. Turney* Submitted to: Artificial Life. Category: Article. Date: June 15, 2020. Abstract We present a computational simulation of evolving entities that includes symbiosis with shifting levels of selection. Evolution by natural selection shifts from the level of the original entities to the level of the new symbiotic entity. In the simulation, the fitness of an entity is measured by a series of one-on-one competitions in the Immigration Game, a two-player variation of Conway’s Game of Life. Mutation, reproduction, and symbiosis are implemented as operations that are external to the Immigration Game. Because these operations are external to the game, we can freely manipulate the operations and observe the effects of the manipulations. The simulation is composed of four layers, each layer building on the previous layer. The first layer implements a simple form of asexual reproduction, the second layer introduces a more sophisticated form of asexual reproduction, the third layer adds sexual reproduction, and the fourth layer adds symbiosis. The experiments show that a small amount of symbiosis, added to the other layers, significantly increases the fitness of the population. We suggest that the model may provide new insights into symbiosis in biological and cultural evolution. Keywords: Symbiosis, cooperation, open-ended evolution, Game of Life, Immigration Game, levels of selection. * Ronin Institute, 127 Haddon Place, Montclair, NJ 07043-2314, USA, [email protected], 819-661-4625. P. D. Turney Symbiosis Promotes Fitness Improvements 1 Introduction There are two main definitions of symbiosis in biology, (1) symbiosis as any association and (2) symbiosis as persistent mutualism [7].
    [Show full text]
  • Programmare Un Sintetizzatore Con Automi Cellulari
    Alma Mater Studiorum UniversitadiBologna` · FACOLTA` DI SCIENZE MATEMATICHE, FISICHE E NATURALI Corso di Laurea in Scienze di Internet Programmare un sintetizzatore con Automi Cellulari Tesi di Laurea in Programmazione Internet Relatore: Presentata da: Chiar.mo Prof. Armando Viliam Fortunato Antonio Messina Sessione III Anno Accademico 2010/2011 Indice (1) Introduzione ...........................................................................................5 1.1 Precedenti ..................................................................................6 (2) Automi Cellulari 2.1 Cosa sono? ................................................................................11 2.2 Come funzionano? .....................................................................13 2.3 Sistemi Dinamici e classi di Automi Cellulari ..............................17 2.4 Computazione Universale e Automi Cellulari ..............................21 2.5 Game of Life ...............................................................................23 - Come funziona .....................................................................23 - Pattern di Stati ....................................................................24 - Oggetti statici e oggetti ripetitivi .........................................25 2.6 Spacefiller e Breeder ...................................................................29 2.7 Supercolliders .............................................................................31 2.8 Auto-riproducibilità ...................................................................33
    [Show full text]
  • The Lifebox, the Seashell, and the Soul
    The Lifebox, the Seashell, and the Soul The Lifebox, the Seashell, and the Soul WHAT GNARLY COMPUTATION TAUGHT ME ABOUT ULTIMATE REALITY, THE MEANING OF LIFE, AND HOW TO BE HAPPY Rudy Rucker THUNDER’S MOUTH PRESS NEW YORK THE LIFEBOX, THE SEASHELL, AND THE SOUL WHAT GNARLY COMPUTATION TAUGHT ME ABOUT ULTIMATE REALITY, THE MEANING OF LIFE, AND HOW TO BE HAPPY Published by Thunder’s Mouth Press An Imprint of Avalon Publishing Group Inc. 245 West 17th St., 11th Floor New York, NY 10011 Copyright © 2005 by Rudy Rucker First printing October 2005 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system now known or to be invented, without permission in writing from the publisher, except by a reviewer who wishes to quote brief passages in connection with a review written for inclusion in a magazine, newspaper, or broadcast. Library of Congress Cataloging-in-Publication Data is available. ISBN 1-56025-722-9 9 8 7 6 5 4 3 2 1 Illustrations by Isabel Rucker Book design by Maria E. Torres Printed in the United States of America Distributed by Publishers Group West Contents PREFACE . ix THOUGHT EXPERIMENT ONE: LUCKY NUMBER . 1 CHAPTER ONE: Computation Everywhere . 5 1.1: Universal Automatism . 5 1.2: A New Kind of Science . 18 1.3: Reckoning a Sum . 31 1.4: Analytical Engines . 36 1.5: The Tiniest Brains . 44 1.6: Inside the Beige Box .
    [Show full text]
  • Cellular Automata Is Λ a Good Measure?
    NAVY Research Group Department of Computer Science Faculty of Electrical Engineering and Computer Science VŠB-TUO 17. listopadu 15 708 33 Ostrava-Poruba Czech Republic! Basics of Modern Computer Science Cellular Automata An Introduction Ivan Zelinka MBCS CIPT, www.bcs.org/ http://www.springer.com/series/10624 Department of Computer Science Faculty of Electrical Engineering and Computer Science, VŠB-TUO 17. listopadu 15 , 708 33 Ostrava-Poruba Czech Republic www.ivanzelinka.eu Topics • Historical overview • Introduction • CA’s and its rules • Wolfram’s classes of CA • Langton’s λ-parameter • CA’s behavior • Patterns • New Kind of Science • CA’s in simulation and modelling 2! navy.cs.vsb.cz! Objectives The objectives of the lesson are: • Explain basics of CA’s • Show that CA’s produce different behavior and patterns • Explain rules of CA and its impact on its dynamics • Show that CA can be used as a pseudorandom number generator • Explain classification of CA’s by Wolfram’s classes 1 – 4 • Demonstrate use of CA’s in modeling and simulations 3! navy.cs.vsb.cz! Cellular Automata Introduction Simple rules can lead to complex behavior. 4! navy.cs.vsb.cz! Cellular Automata Introduction • 40's and 50s: von Neumann, Ulam: basic formalism (the study of self-reproduction) • 1970: The Conway: The Game of Life, an article in Scientific American (Gardner), it attracted big attention • 1983 Wolfram: A review of the CA, CA beginning of study in physics • 1984: Founding of Santa Fe Institute • 1987: The Langton: Artificial Life 5! navy.cs.vsb.cz! Cellular
    [Show full text]
  • Localization Dynamics in a Binary Two-Dimensional Cellular Automaton: the Diffusion Rule
    Localization dynamics in a binary two-dimensional cellular automaton: the Diffusion Rule Genaro J. Mart´ıne1;2,z Andrew Adamatzky2 Harold V. McIntosh3 submitted August 6, 2006 accepted December 2006∗ 1 Departamento de Posgrado, Escuela Superior de C´omputo,Instituto Polit´ecnicoNacional, M´exico. E-mail: [email protected] http://uncomp.uwe.ac.uk/genaro/ 2 Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Bristol, United Kingdom. E-mail: [email protected] http://uncomp.uwe.ac.uk/adamatzky/ 3 Departamento de Aplicaci´onde Microcomputadoras, Instituto de Ciencias, Universidad Aut´onomade Puebla, Puebla, M´exico. E-mail: [email protected] http://delta.cs.cinvestav.mx/~mcintosh/ Abstract We study a two-dimensional cellular automaton (CA), called Diffusion Rule (DR), which exhibits diffusion-like dynamics of propagating patterns. In computational experiments we discover a wide range of mobile and sta- tionary localizations (gliders, oscillators, glider guns, puffer trains, etc), analyze spatio-temporal dynamics of collisions between localizations, and arXiv:0908.0828v1 [cs.FL] 6 Aug 2009 discuss possible applications in unconventional computing. Keywords: Cellular automata, Diffusion Rule, semi-totalistic rules, particle collisions, mean field theory, reaction-diffusion, unconventional computing ∗submitted to Journal of Cellular Automata. 1 1 Introduction In our previous studies on minimal cellular automaton (CA) models of reaction- diffusion chemical system we constructed [4] a binary-cell-state eight-cell neigh- borhood 2D CA model of a quasi-chemical system with one substrate, state 0, and one reagent, state 1. In that model chemical reactions were represented by semi-totalistic transition rules.
    [Show full text]